Daily Papers

Recent studies have drawn attention to the untapped potential of the "star operation" (element-wise multiplication) in network design. While intuitive explanations abound, the foundational rationale behind its application remains largely unexplored. Our study attempts to reveal the star operation's ability to map inputs into high-dimensional, non-linear feature spaces -- akin to kernel tricks -- without widening the network. We further introduce StarNet, a simple yet powerful prototype, demonstrating impressive performance and low latency under compact network structure and efficient budget. Like stars in the sky, the star operation appears unremarkable but holds a vast universe of potential. Our work encourages further exploration across tasks, with codes available at https://github.com/ma-xu/Rewrite-the-Stars.

In-context learning is fundamental to modern Large Language Models (LLMs); however, prevailing architectures impose a rigid and fixed contextual structure by assigning linear or constant positional indices. Drawing on Cognitive Load Theory (CLT), we argue that this uninformative structure increases extraneous cognitive load, consuming finite working memory capacity that should be allocated to deep reasoning and attention allocation. To address this, we propose RePo, a novel mechanism that reduces extraneous load via context re-positioning. Unlike standard approaches, RePo utilizes a differentiable module, f_φ, to assign token positions that capture contextual dependencies, rather than replying on pre-defined integer range. By continually pre-training on the OLMo-2 1B backbone, we demonstrate that RePo significantly enhances performance on tasks involving noisy contexts, structured data, and longer context length, while maintaining competitive performance on general short-context tasks. Detailed analysis reveals that RePo successfully allocate higher attention to distant but relevant information, assign positions in dense and non-linear space, and capture the intrinsic structure of the input context. Our code is available at https://github.com/SakanaAI/repo.

Rotation estimation of high precision from an RGB-D object observation is a huge challenge in 6D object pose estimation, due to the difficulty of learning in the non-linear space of SO(3). In this paper, we propose a novel rotation estimation network, termed as VI-Net, to make the task easier by decoupling the rotation as the combination of a viewpoint rotation and an in-plane rotation. More specifically, VI-Net bases the feature learning on the sphere with two individual branches for the estimates of two factorized rotations, where a V-Branch is employed to learn the viewpoint rotation via binary classification on the spherical signals, while another I-Branch is used to estimate the in-plane rotation by transforming the signals to view from the zenith direction. To process the spherical signals, a Spherical Feature Pyramid Network is constructed based on a novel design of SPAtial Spherical Convolution (SPA-SConv), which settles the boundary problem of spherical signals via feature padding and realizesviewpoint-equivariant feature extraction by symmetric convolutional operations. We apply the proposed VI-Net to the challenging task of category-level 6D object pose estimation for predicting the poses of unknown objects without available CAD models; experiments on the benchmarking datasets confirm the efficacy of our method, which outperforms the existing ones with a large margin in the regime of high precision.

While the recent advances in research on video reenactment have yielded promising results, the approaches fall short in capturing the fine, detailed, and expressive facial features (e.g., lip-pressing, mouth puckering, mouth gaping, and wrinkles) which are crucial in generating realistic animated face videos. To this end, we propose an end-to-end expressive face video encoding approach that facilitates data-efficient high-quality video re-synthesis by optimizing low-dimensional edits of a single Identity-latent. The approach builds on StyleGAN2 image inversion and multi-stage non-linear latent-space editing to generate videos that are nearly comparable to input videos. While existing StyleGAN latent-based editing techniques focus on simply generating plausible edits of static images, we automate the latent-space editing to capture the fine expressive facial deformations in a sequence of frames using an encoding that resides in the Style-latent-space (StyleSpace) of StyleGAN2. The encoding thus obtained could be super-imposed on a single Identity-latent to facilitate re-enactment of face videos at 1024^2. The proposed framework economically captures face identity, head-pose, and complex expressive facial motions at fine levels, and thereby bypasses training, person modeling, dependence on landmarks/ keypoints, and low-resolution synthesis which tend to hamper most re-enactment approaches. The approach is designed with maximum data efficiency, where a single W+ latent and 35 parameters per frame enable high-fidelity video rendering. This pipeline can also be used for puppeteering (i.e., motion transfer).

Non-linear state-space models, also known as general hidden Markov models, are ubiquitous in statistical machine learning, being the most classical generative models for serial data and sequences in general. The particle-based, rapid incremental smoother PaRIS is a sequential Monte Carlo (SMC) technique allowing for efficient online approximation of expectations of additive functionals under the smoothing distribution in these models. Such expectations appear naturally in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing (MSC). PARIS has linear computational complexity, limited memory requirements and comes with non-asymptotic bounds, convergence results and stability guarantees. Still, being based on self-normalised importance sampling, the PaRIS estimator is biased. Our first contribution is to design a novel additive smoothing algorithm, the Parisian particle Gibbs PPG sampler, which can be viewed as a PaRIS algorithm driven by conditional SMC moves, resulting in bias-reduced estimates of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities. Our second contribution is to apply PPG in a learning framework, covering MLE and MSC as special examples. In this context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value of bias reduction and the implicit Rao--Blackwellization of PPG. These are the first non-asymptotic results of this kind in this setting. We illustrate our theoretical results with numerical experiments supporting our claims.

Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.

The Linear Representation Hypothesis (LRH) states that neural networks learn to encode concepts as directions in activation space, and a strong version of the LRH states that models learn only such encodings. In this paper, we present a counterexample to this strong LRH: when trained to repeat an input token sequence, gated recurrent neural networks (RNNs) learn to represent the token at each position with a particular order of magnitude, rather than a direction. These representations have layered features that are impossible to locate in distinct linear subspaces. To show this, we train interventions to predict and manipulate tokens by learning the scaling factor corresponding to each sequence position. These interventions indicate that the smallest RNNs find only this magnitude-based solution, while larger RNNs have linear representations. These findings strongly indicate that interpretability research should not be confined by the LRH.

Popular guidance for denoising diffusion probabilistic model (DDPM) linearly combines distinct conditional models together to provide enhanced control over samples. However, this approach overlooks nonlinear effects that become significant when guidance scale is large. To address this issue, we propose characteristic guidance, a guidance method that provides first-principle non-linear correction for classifier-free guidance. Such correction forces the guided DDPMs to respect the Fokker-Planck (FP) equation of diffusion process, in a way that is training-free and compatible with existing sampling methods. Experiments show that characteristic guidance enhances semantic characteristics of prompts and mitigate irregularities in image generation, proving effective in diverse applications ranging from simulating magnet phase transitions to latent space sampling.

Large Language Models (LLMs) were shown to struggle with long-term planning, which may be caused by the limited way in which they explore the space of possible solutions. We propose an architecture where a Reinforcement Learning (RL) Agent guides an LLM's space exploration: (1) the Agent has access to domain-specific information, and can therefore make decisions about the quality of candidate solutions based on specific and relevant metrics, which were not explicitly considered by the LLM's training objective; (2) the LLM can focus on generating immediate next steps, without the need for long-term planning. We allow non-linear reasoning by exploring alternative paths and backtracking. We evaluate this architecture on the program equivalence task, and compare it against Chain of Thought (CoT) and Tree of Thoughts (ToT). We assess both the downstream task, denoting the binary classification, and the intermediate reasoning steps. Our approach compares positively against CoT and ToT.

Reasoning from sequences of raw sensory data is a ubiquitous problem across fields ranging from medical devices to robotics. These problems often involve using long sequences of raw sensor data (e.g. magnetometers, piezoresistors) to predict sequences of desirable physical quantities (e.g. force, inertial measurements). While classical approaches are powerful for locally-linear prediction problems, they often fall short when using real-world sensors. These sensors are typically non-linear, are affected by extraneous variables (e.g. vibration), and exhibit data-dependent drift. For many problems, the prediction task is exacerbated by small labeled datasets since obtaining ground-truth labels requires expensive equipment. In this work, we present Hierarchical State-Space Models (HiSS), a conceptually simple, new technique for continuous sequential prediction. HiSS stacks structured state-space models on top of each other to create a temporal hierarchy. Across six real-world sensor datasets, from tactile-based state prediction to accelerometer-based inertial measurement, HiSS outperforms state-of-the-art sequence models such as causal Transformers, LSTMs, S4, and Mamba by at least 23% on MSE. Our experiments further indicate that HiSS demonstrates efficient scaling to smaller datasets and is compatible with existing data-filtering techniques. Code, datasets and videos can be found on https://hiss-csp.github.io.

Existing works have extensively studied adversarial examples, which are minimal perturbations that can mislead the output of deep neural networks (DNNs) while remaining imperceptible to humans. However, in this work, we reveal the existence of a harmless perturbation space, in which perturbations drawn from this space, regardless of their magnitudes, leave the network output unchanged when applied to inputs. Essentially, the harmless perturbation space emerges from the usage of non-injective functions (linear or non-linear layers) within DNNs, enabling multiple distinct inputs to be mapped to the same output. For linear layers with input dimensions exceeding output dimensions, any linear combination of the orthogonal bases of the nullspace of the parameter consistently yields no change in their output. For non-linear layers, the harmless perturbation space may expand, depending on the properties of the layers and input samples. Inspired by this property of DNNs, we solve for a family of general perturbation spaces that are redundant for the DNN's decision, and can be used to hide sensitive data and serve as a means of model identification. Our work highlights the distinctive robustness of DNNs (i.e., consistency under large magnitude perturbations) in contrast to adversarial examples (vulnerability for small imperceptible noises).

In this article, we consider a newly proposed parameterization of the viscosity coefficient zeta, specifically zeta=zeta_0 {Omega^s_m} H , where zeta_0 = zeta_0{{Omega^s_{m_0}}} within the coincident f(Q) gravity formalism. We consider a non-linear function f(Q)= -Q +alpha Q^n, where alpha and n are arbitrary model parameters, which is a power-law correction to the STEGR scenario. We find an autonomous system by invoking the dimensionless density parameters as the governing phase-space variables. We discuss the physical significance of the model corresponding to the parameter choices n=-1 and n=2 along with the exponent choices s=0, 0.5, and 1.05. We find that model I shows the stable de-Sitter type or stable phantom type (depending on the choice of exponent s) behavior with no transition epoch, whereas model II shows the evolutionary phase from the radiation epoch to the accelerated de-Sitter epoch via passing through the matter-dominated epoch. Hence, we conclude that model I provides a good description of the late-time cosmology but fails to describe the transition epoch, whereas model II modifies the description in the context of the early universe and provides a good description of the matter and radiation era along with the transition phase.

Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Modeling multivariate time series is a well-established problem with a wide range of applications from healthcare to financial markets. Traditional State Space Models (SSMs) are classical approaches for univariate time series modeling due to their simplicity and expressive power to represent linear dependencies. They, however, have fundamentally limited expressive power to capture non-linear dependencies, are slow in practice, and fail to model the inter-variate information flow. Despite recent attempts to improve the expressive power of SSMs by using deep structured SSMs, the existing methods are either limited to univariate time series, fail to model complex patterns (e.g., seasonal patterns), fail to dynamically model the dependencies of variate and time dimensions, and/or are input-independent. We present Chimera that uses two input-dependent 2-D SSM heads with different discretization processes to learn long-term progression and seasonal patterns. To improve the efficiency of complex 2D recurrence, we present a fast training using a new 2-dimensional parallel selective scan. We further present and discuss 2-dimensional Mamba and Mamba-2 as the spacial cases of our 2D SSM. Our experimental evaluation shows the superior performance of Chimera on extensive and diverse benchmarks, including ECG and speech time series classification, long-term and short-term time series forecasting, and time series anomaly detection.

In this study we explore the optimization of laser pulse duration to obtain the shortest possible pulse. We do this by employing a feedback loop between a pulse shaper and pulse duration measurements. We apply to this problem several iterative algorithms including gradient descent, Bayesian Optimization and genetic algorithms, using a simulation of the actual laser represented via a semi-physical model of the laser based on the process of linear and non-linear phase accumulation.

Automated analysis of complex systems based on multiple readouts remains a challenge. Change point detection algorithms are aimed to locating abrupt changes in the time series behaviour of a process. In this paper, we present a novel change point detection algorithm based on Latent Neural Stochastic Differential Equations (SDE). Our method learns a non-linear deep learning transformation of the process into a latent space and estimates a SDE that describes its evolution over time. The algorithm uses the likelihood ratio of the learned stochastic processes in different timestamps to find change points of the process. We demonstrate the detection capabilities and performance of our algorithm on synthetic and real-world datasets. The proposed method outperforms the state-of-the-art algorithms on the majority of our experiments.

Flow-based latent generative models such as Stable Diffusion 3 are able to generate images with remarkable quality, even enabling photorealistic text-to-image generation. Their impressive performance suggests that these models should also constitute powerful priors for inverse imaging problems, but that approach has not yet led to comparable fidelity. There are several key obstacles: (i) the encoding into a lower-dimensional latent space makes the underlying (forward) mapping non-linear; (ii) the data likelihood term is usually intractable; and (iii) learned generative models struggle to recover rare, atypical data modes during inference. We present FLAIR, a novel training free variational framework that leverages flow-based generative models as a prior for inverse problems. To that end, we introduce a variational objective for flow matching that is agnostic to the type of degradation, and combine it with deterministic trajectory adjustments to recover atypical modes. To enforce exact consistency with the observed data, we decouple the optimization of the data fidelity and regularization terms. Moreover, we introduce a time-dependent calibration scheme in which the strength of the regularization is modulated according to off-line accuracy estimates. Results on standard imaging benchmarks demonstrate that FLAIR consistently outperforms existing diffusion- and flow-based methods in terms of reconstruction quality and sample diversity.

We introduce Z-SASLM, a Zero-Shot Style-Aligned SLI (Spherical Linear Interpolation) Blending Latent Manipulation pipeline that overcomes the limitations of current multi-style blending methods. Conventional approaches rely on linear blending, assuming a flat latent space leading to suboptimal results when integrating multiple reference styles. In contrast, our framework leverages the non-linear geometry of the latent space by using SLI Blending to combine weighted style representations. By interpolating along the geodesic on the hypersphere, Z-SASLM preserves the intrinsic structure of the latent space, ensuring high-fidelity and coherent blending of diverse styles - all without the need for fine-tuning. We further propose a new metric, Weighted Multi-Style DINO ViT-B/8, designed to quantitatively evaluate the consistency of the blended styles. While our primary focus is on the theoretical and practical advantages of SLI Blending for style manipulation, we also demonstrate its effectiveness in a multi-modal content fusion setting through comprehensive experimental studies. Experimental results show that Z-SASLM achieves enhanced and robust style alignment. The implementation code can be found at: https://github.com/alessioborgi/Z-SASLM.

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a distributed universal approximation theorem guaranteeing that any Lipschitz non-linear operator between L^2([0,1]^d) spaces can be approximated uniformly over the Sobolev unit ball therein, to any given varepsilon>0 accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of O(varepsilon^{-1}). Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies varepsilon. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of L^2([0,1]^d).

Reconstructions of visual perception from brain activity have improved tremendously, but the practical utility of such methods has been limited. This is because such models are trained independently per subject where each subject requires dozens of hours of expensive fMRI training data to attain high-quality results. The present work showcases high-quality reconstructions using only 1 hour of fMRI training data. We pretrain our model across 7 subjects and then fine-tune on minimal data from a new subject. Our novel functional alignment procedure linearly maps all brain data to a shared-subject latent space, followed by a shared non-linear mapping to CLIP image space. We then map from CLIP space to pixel space by fine-tuning Stable Diffusion XL to accept CLIP latents as inputs instead of text. This approach improves out-of-subject generalization with limited training data and also attains state-of-the-art image retrieval and reconstruction metrics compared to single-subject approaches. MindEye2 demonstrates how accurate reconstructions of perception are possible from a single visit to the MRI facility. All code is available on GitHub.

Reconstructing 3D human shape and pose from monocular images is challenging despite the promising results achieved by the most recent learning-based methods. The commonly occurred misalignment comes from the facts that the mapping from images to the model space is highly non-linear and the rotation-based pose representation of body models is prone to result in the drift of joint positions. In this work, we investigate learning 3D human shape and pose from dense correspondences of body parts and propose a Decompose-and-aggregate Network (DaNet) to address these issues. DaNet adopts the dense correspondence maps, which densely build a bridge between 2D pixels and 3D vertices, as intermediate representations to facilitate the learning of 2D-to-3D mapping. The prediction modules of DaNet are decomposed into one global stream and multiple local streams to enable global and fine-grained perceptions for the shape and pose predictions, respectively. Messages from local streams are further aggregated to enhance the robust prediction of the rotation-based poses, where a position-aided rotation feature refinement strategy is proposed to exploit spatial relationships between body joints. Moreover, a Part-based Dropout (PartDrop) strategy is introduced to drop out dense information from intermediate representations during training, encouraging the network to focus on more complementary body parts as well as neighboring position features. The efficacy of the proposed method is validated on both indoor and real-world datasets including Human3.6M, UP3D, COCO, and 3DPW, showing that our method could significantly improve the reconstruction performance in comparison with previous state-of-the-art methods. Our code is publicly available at https://hongwenzhang.github.io/dense2mesh .

Auditing large language models for unexpected behaviors is critical to preempt catastrophic deployments, yet remains challenging. In this work, we cast auditing as an optimization problem, where we automatically search for input-output pairs that match a desired target behavior. For example, we might aim to find a non-toxic input that starts with "Barack Obama" that a model maps to a toxic output. This optimization problem is difficult to solve as the set of feasible points is sparse, the space is discrete, and the language models we audit are non-linear and high-dimensional. To combat these challenges, we introduce a discrete optimization algorithm, ARCA, that jointly and efficiently optimizes over inputs and outputs. Our approach automatically uncovers derogatory completions about celebrities (e.g. "Barack Obama is a legalized unborn" -> "child murderer"), produces French inputs that complete to English outputs, and finds inputs that generate a specific name. Our work offers a promising new tool to uncover models' failure-modes before deployment.

The task of age transformation illustrates the change of an individual's appearance over time. Accurately modeling this complex transformation over an input facial image is extremely challenging as it requires making convincing, possibly large changes to facial features and head shape, while still preserving the input identity. In this work, we present an image-to-image translation method that learns to directly encode real facial images into the latent space of a pre-trained unconditional GAN (e.g., StyleGAN) subject to a given aging shift. We employ a pre-trained age regression network to explicitly guide the encoder in generating the latent codes corresponding to the desired age. In this formulation, our method approaches the continuous aging process as a regression task between the input age and desired target age, providing fine-grained control over the generated image. Moreover, unlike approaches that operate solely in the latent space using a prior on the path controlling age, our method learns a more disentangled, non-linear path. Finally, we demonstrate that the end-to-end nature of our approach, coupled with the rich semantic latent space of StyleGAN, allows for further editing of the generated images. Qualitative and quantitative evaluations show the advantages of our method compared to state-of-the-art approaches.

The solar wind is a medium characterized by strong turbulence and significant field fluctuations on various scales. Recent observations have revealed that magnetic turbulence exhibits a self-similar behavior. Similarly, high-resolution measurements of the proton density have shown comparable characteristics, prompting several studies into the multifractal properties of these density fluctuations. In this work, we show that low-resolution observations of the solar wind proton density over time, recorded by various spacecraft at Lagrange point L1, also exhibit non-linear and multifractal structures. The novelty of our study lies in the fact that this is the first systematic analysis of solar wind proton density using low-resolution (hourly) data collected by multiple spacecraft at the L1 Lagrange point over a span of 17 years. Furthermore, we interpret our results within the framework of non-extensive statistical mechanics, which appears to be consistent with the observed nonlinear behavior. Based on the data, we successfully validate the q-triplet predicted by non-extensive statistical theory. To the best of our knowledge, this represents the most rigorous and systematic validation to date of the q-triplet in the solar wind.

We provide a concise framework for generalized ensemble theory through a matrix-based approach. By introducing an observation matrix, any discrete probability distribution, including those for non-equilibrium steady states, can be expressed as a generalized Boltzmann distribution, with observables and conjugate variables as the basis and coordinates in a linear space. In this framework, we identify the minimal sufficient statistics required for inferring the Boltzmann distribution. Furthermore, we show that the Hadamard and Vandermonde matrices are suitable observation matrices for spin systems and random walks. In master equation systems, the probability flux observation matrix facilitates the identification of detailed balance violations. Our findings provide a new approach to developing generalized ensemble theory for non-equilibrium steady-state systems.

We consider the problem of how a teacher algorithm can enable an unknown Deep Reinforcement Learning (DRL) student to become good at a skill over a wide range of diverse environments. To do so, we study how a teacher algorithm can learn to generate a learning curriculum, whereby it sequentially samples parameters controlling a stochastic procedural generation of environments. Because it does not initially know the capacities of its student, a key challenge for the teacher is to discover which environments are easy, difficult or unlearnable, and in what order to propose them to maximize the efficiency of learning over the learnable ones. To achieve this, this problem is transformed into a surrogate continuous bandit problem where the teacher samples environments in order to maximize absolute learning progress of its student. We present a new algorithm modeling absolute learning progress with Gaussian mixture models (ALP-GMM). We also adapt existing algorithms and provide a complete study in the context of DRL. Using parameterized variants of the BipedalWalker environment, we study their efficiency to personalize a learning curriculum for different learners (embodiments), their robustness to the ratio of learnable/unlearnable environments, and their scalability to non-linear and high-dimensional parameter spaces. Videos and code are available at https://github.com/flowersteam/teachDeepRL.

Recent years have witnessed significant progress in generative models for music, featuring diverse architectures that balance output quality, diversity, speed, and user control. This study explores a user-friendly graphical interface enabling the drawing of masked regions for inpainting by an Hourglass Diffusion Transformer (HDiT) model trained on MIDI piano roll images. To enhance note generation in specified areas, masked regions can be "repainted" with extra noise. The non-latent HDiTs linear scaling with pixel count allows efficient generation in pixel space, providing intuitive and interpretable controls such as masking throughout the network and removing the need to operate in compressed latent spaces such as those provided by pretrained autoencoders. We demonstrate that, in addition to inpainting of melodies, accompaniment, and continuations, the use of repainting can help increase note density yielding musical structures closely matching user specifications such as rising, falling, or diverging melody and/or accompaniment, even when these lie outside the typical training data distribution. We achieve performance on par with prior results while operating at longer context windows, with no autoencoder, and can enable complex geometries for inpainting masks, increasing the options for machine-assisted composers to control the generated music.

Vision transformers have significantly advanced the field of computer vision, offering robust modeling capabilities and global receptive field. However, their high computational demands limit their applicability in processing long sequences. To tackle this issue, State Space Models (SSMs) have gained prominence in vision tasks as they offer linear computational complexity. Recently, State Space Duality (SSD), an improved variant of SSMs, was introduced in Mamba2 to enhance model performance and efficiency. However, the inherent causal nature of SSD/SSMs restricts their applications in non-causal vision tasks. To address this limitation, we introduce Visual State Space Duality (VSSD) model, which has a non-causal format of SSD. Specifically, we propose to discard the magnitude of interactions between the hidden state and tokens while preserving their relative weights, which relieves the dependencies of token contribution on previous tokens. Together with the involvement of multi-scan strategies, we show that the scanning results can be integrated to achieve non-causality, which not only improves the performance of SSD in vision tasks but also enhances its efficiency. We conduct extensive experiments on various benchmarks including image classification, detection, and segmentation, where VSSD surpasses existing state-of-the-art SSM-based models. Code and weights are available at https://github.com/YuHengsss/VSSD.

The optimal transport problem for measures supported on non-Euclidean spaces has recently gained ample interest in diverse applications involving representation learning. In this paper, we focus on circular probability measures, i.e., probability measures supported on the unit circle, and introduce a new computationally efficient metric for these measures, denoted as Linear Circular Optimal Transport (LCOT). The proposed metric comes with an explicit linear embedding that allows one to apply Machine Learning (ML) algorithms to the embedded measures and seamlessly modify the underlying metric for the ML algorithm to LCOT. We show that the proposed metric is rooted in the Circular Optimal Transport (COT) and can be considered the linearization of the COT metric with respect to a fixed reference measure. We provide a theoretical analysis of the proposed metric and derive the computational complexities for pairwise comparison of circular probability measures. Lastly, through a set of numerical experiments, we demonstrate the benefits of LCOT in learning representations of circular measures.

Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, f:XtoY. Is it possible to identify a pair of non-standard vector spaces for which a conventionally nonlinear function is, in fact, linear? This paper introduces a method that makes such vector spaces explicit by construction. We find that if we sandwich a linear operator A between two invertible neural networks, f(x)=g_y^{-1}(A g_x(x)), then the corresponding vector spaces X and Y are induced by newly defined addition and scaling actions derived from g_x and g_y. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. f(f(x))=f(x)) on networks leading to a globally projective generative model and to demonstrate modular style transfer.

Large Language Models (LLMs) increasingly incorporate multilingual capabilities, fueling the demand to transfer them into target language-specific models. However, most approaches, which blend the source model's embedding by replacing the source vocabulary with the target language-specific vocabulary, may constrain expressive capacity in the target language since the source model is predominantly trained on English data. In this paper, we propose Semantic Aware Linear Transfer (SALT), a novel cross-lingual transfer technique that recycles embeddings from target language Pre-trained Language Models (PLMs) to transmit the deep representational strengths of PLM-derived embedding to LLMs. SALT derives unique regression lines based on the similarity in the overlap of the source and target vocabularies, to handle each non-overlapping token's embedding space. Our extensive experiments show that SALT significantly outperforms other transfer methods and achieves lower loss with accelerating faster convergence during language adaptation. Notably, SALT obtains remarkable performance in cross-lingual understanding setups compared to other methods. Furthermore, we highlight the scalable use of PLMs to enhance the functionality of contemporary LLMs by conducting experiments with varying architectures.

In Online Continual Learning (OCL) a learning system receives a stream of data and sequentially performs prediction and training steps. Important challenges in OCL are concerned with automatic adaptation to the particular non-stationary structure of the data, and with quantification of predictive uncertainty. Motivated by these challenges we introduce a probabilistic Bayesian online learning model by using a (possibly pretrained) neural representation and a state space model over the linear predictor weights. Non-stationarity over the linear predictor weights is modelled using a parameter drift transition density, parametrized by a coefficient that quantifies forgetting. Inference in the model is implemented with efficient Kalman filter recursions which track the posterior distribution over the linear weights, while online SGD updates over the transition dynamics coefficient allows to adapt to the non-stationarity seen in data. While the framework is developed assuming a linear Gaussian model, we also extend it to deal with classification problems and for fine-tuning the deep learning representation. In a set of experiments in multi-class classification using data sets such as CIFAR-100 and CLOC we demonstrate the predictive ability of the model and its flexibility to capture non-stationarity.

In this paper, we revisit the problem of sparse linear regression in the local differential privacy (LDP) model. Existing research in the non-interactive and sequentially local models has focused on obtaining the lower bounds for the case where the underlying parameter is 1-sparse, and extending such bounds to the more general k-sparse case has proven to be challenging. Moreover, it is unclear whether efficient non-interactive LDP (NLDP) algorithms exist. To address these issues, we first consider the problem in the epsilon non-interactive LDP model and provide a lower bound of Omega(sqrt{dklog d}{nepsilon}) on the ell_2-norm estimation error for sub-Gaussian data, where n is the sample size and d is the dimension of the space. We propose an innovative NLDP algorithm, the very first of its kind for the problem. As a remarkable outcome, this algorithm also yields a novel and highly efficient estimator as a valuable by-product. Our algorithm achieves an upper bound of O({dsqrt{k}{nepsilon}}) for the estimation error when the data is sub-Gaussian, which can be further improved by a factor of O(d) if the server has additional public but unlabeled data. For the sequentially interactive LDP model, we show a similar lower bound of Omega({sqrt{dk}{nepsilon}}). As for the upper bound, we rectify a previous method and show that it is possible to achieve a bound of O(ksqrt{d}{nepsilon}). Our findings reveal fundamental differences between the non-private case, central DP model, and local DP model in the sparse linear regression problem.

Gating mechanisms have been widely utilized, from early models like LSTMs and Highway Networks to recent state space models, linear attention, and also softmax attention. Yet, existing literature rarely examines the specific effects of gating. In this work, we conduct comprehensive experiments to systematically investigate gating-augmented softmax attention variants. Specifically, we perform a comprehensive comparison over 30 variants of 15B Mixture-of-Experts (MoE) models and 1.7B dense models trained on a 3.5 trillion token dataset. Our central finding is that a simple modification-applying a head-specific sigmoid gate after the Scaled Dot-Product Attention (SDPA)-consistently improves performance. This modification also enhances training stability, tolerates larger learning rates, and improves scaling properties. By comparing various gating positions and computational variants, we attribute this effectiveness to two key factors: (1) introducing non-linearity upon the low-rank mapping in the softmax attention, and (2) applying query-dependent sparse gating scores to modulate the SDPA output. Notably, we find this sparse gating mechanism mitigates 'attention sink' and enhances long-context extrapolation performance, and we also release related https://github.com/qiuzh20/gated_attention{codes} and https://huggingface.co/QwQZh/gated_attention{models} to facilitate future research.

To mitigate the computational complexity in the self-attention mechanism on long sequences, linear attention utilizes computation tricks to achieve linear complexity, while state space models (SSMs) popularize a favorable practice of using non-data-dependent memory pattern, i.e., emphasize the near and neglect the distant, to processing sequences. Recent studies have shown the priorities by combining them as one. However, the efficiency of linear attention remains only at the theoretical level in a causal setting, and SSMs require various designed constraints to operate effectively on specific data. Therefore, in order to unveil the true power of the hybrid design, the following two issues need to be addressed: (1) hardware-efficient implementation for linear attention and (2) stabilization of SSMs. To achieve this, we leverage the thought of tiling and hierarchy to propose CHELA (short-long Convolutions with Hardware-Efficient Linear Attention), which replaces SSMs with short-long convolutions and implements linear attention in a divide-and-conquer manner. This approach enjoys global abstraction and data-dependent selection from stable SSM and linear attention while maintaining real linear complexity. Our comprehensive experiments on the Long Range Arena benchmark and language modeling tasks demonstrate the effectiveness of the proposed method.

We present Backpacks: a new neural architecture that marries strong modeling performance with an interface for interpretability and control. Backpacks learn multiple non-contextual sense vectors for each word in a vocabulary, and represent a word in a sequence as a context-dependent, non-negative linear combination of sense vectors in this sequence. We find that, after training, sense vectors specialize, each encoding a different aspect of a word. We can interpret a sense vector by inspecting its (non-contextual, linear) projection onto the output space, and intervene on these interpretable hooks to change the model's behavior in predictable ways. We train a 170M-parameter Backpack language model on OpenWebText, matching the loss of a GPT-2 small (124Mparameter) Transformer. On lexical similarity evaluations, we find that Backpack sense vectors outperform even a 6B-parameter Transformer LM's word embeddings. Finally, we present simple algorithms that intervene on sense vectors to perform controllable text generation and debiasing. For example, we can edit the sense vocabulary to tend more towards a topic, or localize a source of gender bias to a sense vector and globally suppress that sense.

While multitask representation learning has become a popular approach in reinforcement learning (RL) to boost the sample efficiency, the theoretical understanding of why and how it works is still limited. Most previous analytical works could only assume that the representation function is already known to the agent or from linear function class, since analyzing general function class representation encounters non-trivial technical obstacles such as generalization guarantee, formulation of confidence bound in abstract function space, etc. However, linear-case analysis heavily relies on the particularity of linear function class, while real-world practice usually adopts general non-linear representation functions like neural networks. This significantly reduces its applicability. In this work, we extend the analysis to general function class representations. Specifically, we consider an agent playing M contextual bandits (or MDPs) concurrently and extracting a shared representation function phi from a specific function class Phi using our proposed Generalized Functional Upper Confidence Bound algorithm (GFUCB). We theoretically validate the benefit of multitask representation learning within general function class for bandits and linear MDP for the first time. Lastly, we conduct experiments to demonstrate the effectiveness of our algorithm with neural net representation.

State space models (SSM) have been widely applied for the analysis and visualization of large sequential datasets. Sequential Monte Carlo (SMC) is a very popular particle-based method to sample latent states from intractable posteriors. However, SSM is significantly influenced by the choice of the proposal. Recently Hamiltonian Monte Carlo (HMC) sampling has shown success in many practical problems. In this paper, we propose an SMC augmented by HMC (HSMC) for inference and model learning of nonlinear SSM, which can exempt us from learning proposals and reduce the model complexity significantly. Based on the measure preserving property of HMC, the particles directly generated by transition function can approximate the posterior of latent states arbitrarily well. In order to better adapt to the local geometry of latent space, the HMC is conducted on Riemannian manifold defined by a positive definite metric. In addition, we show that the proposed HSMC method can improve SSMs realized by both Gaussian Processes (GP) and Neural Network (NN).

We develop a novel policy synthesis algorithm, RMPflow, based on geometrically consistent transformations of Riemannian Motion Policies (RMPs). RMPs are a class of reactive motion policies designed to parameterize non-Euclidean behaviors as dynamical systems in intrinsically nonlinear task spaces. Given a set of RMPs designed for individual tasks, RMPflow can consistently combine these local policies to generate an expressive global policy, while simultaneously exploiting sparse structure for computational efficiency. We study the geometric properties of RMPflow and provide sufficient conditions for stability. Finally, we experimentally demonstrate that accounting for the geometry of task policies can simplify classically difficult problems, such as planning through clutter on high-DOF manipulation systems.

Appropriately designing the proposal kernel of particle filters is an issue of significant importance, since a bad choice may lead to deterioration of the particle sample and, consequently, waste of computational power. In this paper we introduce a novel algorithm adaptively approximating the so-called optimal proposal kernel by a mixture of integrated curved exponential distributions with logistic weights. This family of distributions, referred to as mixtures of experts, is broad enough to be used in the presence of multi-modality or strongly skewed distributions. The mixtures are fitted, via online-EM methods, to the optimal kernel through minimisation of the Kullback-Leibler divergence between the auxiliary target and instrumental distributions of the particle filter. At each iteration of the particle filter, the algorithm is required to solve only a single optimisation problem for the whole particle sample, yielding an algorithm with only linear complexity. In addition, we illustrate in a simulation study how the method can be successfully applied to optimal filtering in nonlinear state-space models.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely high-dimensional combinatorial and nonlinear hypothesis spaces. Traditional methods of equation discovery largely focus on extracting equations from data alone, often neglecting the rich domain-specific prior knowledge that scientists typically depend on. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data in an efficient manner. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeletons, drawing from its physical understanding, which are then optimized against data to estimate skeleton parameters. We demonstrate LLM-SR's effectiveness across three diverse scientific domains, where it discovers physically accurate equations that provide significantly better fits to in-domain and out-of-domain data compared to the well-established equation discovery baselines

Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding. Traditional methods, such as principal component analysis (PCA) and autoencoders (AE), operate in an unsupervised manner, ignoring label information even when it is available. In this work, we introduce a unified method capable of learning latent spaces in both unsupervised and supervised settings. We formulate the problem as a nonlinear multiple-response regression within an index model context. By applying the generalized Stein's lemma, the latent space can be estimated without knowing the nonlinear link functions. Our method can be viewed as a nonlinear generalization of PCA. Moreover, unlike AE and other neural network methods that operate as "black boxes", our approach not only offers better interpretability but also reduces computational complexity while providing strong theoretical guarantees. Comprehensive numerical experiments and real data analyses demonstrate the superior performance of our method.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an encoder, which maps a high-dimensional dataset and its low-dimensional embedding, and a decoder, which takes the embedded data back to the high-dimensional space. Stacking the encoder and decoder together constructs an autoencoder, which we term a diffusion net, that performs out-of-sample-extension as well as outlier detection. We introduce new neural net constraints for the encoder, which preserves the local geometry of the points, and we prove rates of convergence for the encoder. Also, our approach is efficient in both computational complexity and memory requirements, as opposed to previous methods that require storage of all training points in both the high-dimensional and the low-dimensional spaces to calculate the out-of-sample-extension and the pre-image.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

Recent works for face editing usually manipulate the latent space of StyleGAN via the linear semantic directions. However, they usually suffer from the entanglement of facial attributes, need to tune the optimal editing strength, and are limited to binary attributes with strong supervision signals. This paper proposes a novel adaptive nonlinear latent transformation for disentangled and conditional face editing, termed AdaTrans. Specifically, our AdaTrans divides the manipulation process into several finer steps; i.e., the direction and size at each step are conditioned on both the facial attributes and the latent codes. In this way, AdaTrans describes an adaptive nonlinear transformation trajectory to manipulate the faces into target attributes while keeping other attributes unchanged. Then, AdaTrans leverages a predefined density model to constrain the learned trajectory in the distribution of latent codes by maximizing the likelihood of transformed latent code. Moreover, we also propose a disentangled learning strategy under a mutual information framework to eliminate the entanglement among attributes, which can further relax the need for labeled data. Consequently, AdaTrans enables a controllable face editing with the advantages of disentanglement, flexibility with non-binary attributes, and high fidelity. Extensive experimental results on various facial attributes demonstrate the qualitative and quantitative effectiveness of the proposed AdaTrans over existing state-of-the-art methods, especially in the most challenging scenarios with a large age gap and few labeled examples. The source code is available at https://github.com/Hzzone/AdaTrans.

Generalized Category Discovery (GCD) is an intriguing open-world problem that has garnered increasing attention. Given a dataset that includes both labelled and unlabelled images, GCD aims to categorize all images in the unlabelled subset, regardless of whether they belong to known or unknown classes. In GCD, the common practice typically involves applying a spherical projection operator at the end of the self-supervised pretrained backbone, operating within Euclidean or spherical space. However, both of these spaces have been shown to be suboptimal for encoding samples that possesses hierarchical structures. In contrast, hyperbolic space exhibits exponential volume growth relative to radius, making it inherently strong at capturing the hierarchical structure of samples from both seen and unseen categories. Therefore, we propose to tackle the category discovery challenge in the hyperbolic space. We introduce HypCD, a simple Hyperbolic framework for learning hierarchy-aware representations and classifiers for generalized Category Discovery. HypCD first transforms the Euclidean embedding space of the backbone network into hyperbolic space, facilitating subsequent representation and classification learning by considering both hyperbolic distance and the angle between samples. This approach is particularly helpful for knowledge transfer from known to unknown categories in GCD. We thoroughly evaluate HypCD on public GCD benchmarks, by applying it to various baseline and state-of-the-art methods, consistently achieving significant improvements.

This paper presents MetricGrids, a novel grid-based neural representation that combines elementary metric grids in various metric spaces to approximate complex nonlinear signals. While grid-based representations are widely adopted for their efficiency and scalability, the existing feature grids with linear indexing for continuous-space points can only provide degenerate linear latent space representations, and such representations cannot be adequately compensated to represent complex nonlinear signals by the following compact decoder. To address this problem while keeping the simplicity of a regular grid structure, our approach builds upon the standard grid-based paradigm by constructing multiple elementary metric grids as high-order terms to approximate complex nonlinearities, following the Taylor expansion principle. Furthermore, we enhance model compactness with hash encoding based on different sparsities of the grids to prevent detrimental hash collisions, and a high-order extrapolation decoder to reduce explicit grid storage requirements. experimental results on both 2D and 3D reconstructions demonstrate the superior fitting and rendering accuracy of the proposed method across diverse signal types, validating its robustness and generalizability. Code is available at https://github.com/wangshu31/MetricGrids}{https://github.com/wangshu31/MetricGrids.

Through evolution, nature has presented a set of remarkable protein materials, including elastins, silks, keratins and collagens with superior mechanical performances that play crucial roles in mechanobiology. However, going beyond natural designs to discover proteins that meet specified mechanical properties remains challenging. Here we report a generative model that predicts protein designs to meet complex nonlinear mechanical property-design objectives. Our model leverages deep knowledge on protein sequences from a pre-trained protein language model and maps mechanical unfolding responses to create novel proteins. Via full-atom molecular simulations for direct validation, we demonstrate that the designed proteins are novel, and fulfill the targeted mechanical properties, including unfolding energy and mechanical strength, as well as the detailed unfolding force-separation curves. Our model offers rapid pathways to explore the enormous mechanobiological protein sequence space unconstrained by biological synthesis, using mechanical features as target to enable the discovery of protein materials with superior mechanical properties.

We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces properties of a target function under the assumption that it can be effectively approximated by a hypothesis space. In particular, we show that nonlinear sequence relationships that can be stably approximated by nonlinear RNNs must have an exponential decaying memory structure - a notion that can be made precise. This extends the previously identified curse of memory in linear RNNs into the general nonlinear setting, and quantifies the essential limitations of the RNN architecture for learning sequential relationships with long-term memory. Based on the analysis, we propose a principled reparameterization method to overcome the limitations. Our theoretical results are confirmed by numerical experiments. The code has been released in https://github.com/radarFudan/Curse-of-memory

Recurrent Neural Networks (RNNs) laid the foundation for sequence modeling, but their intrinsic sequential nature restricts parallel computation, creating a fundamental barrier to scaling. This has led to the dominance of parallelizable architectures like Transformers and, more recently, State Space Models (SSMs). While SSMs achieve efficient parallelization through structured linear recurrences, this linearity constraint limits their expressive power and precludes modeling complex, nonlinear sequence-wise dependencies. To address this, we present ParaRNN, a framework that breaks the sequence-parallelization barrier for nonlinear RNNs. Building on prior work, we cast the sequence of nonlinear recurrence relationships as a single system of equations, which we solve in parallel using Newton's iterations combined with custom parallel reductions. Our implementation achieves speedups of up to 665x over naive sequential application, allowing training nonlinear RNNs at unprecedented scales. To showcase this, we apply ParaRNN to adaptations of LSTM and GRU architectures, successfully training models of 7B parameters that attain perplexity comparable to similarly-sized Transformers and Mamba2 architectures. To accelerate research in efficient sequence modeling, we release the ParaRNN codebase as an open-source framework for automatic training-parallelization of nonlinear RNNs, enabling researchers and practitioners to explore new nonlinear RNN models at scale.

We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. HOO2024, which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

We present an open-source Python implementation of an idealized high-order pseudo-spectral solver for the one-dimensional nonlinear Schr\"odinger equation (NLSE). The solver combines Fourier spectral spatial discretization with an adaptive eighth-order Dormand-Prince time integration scheme to achieve machine-precision conservation of mass and near-perfect preservation of momentum and energy for smooth solutions. The implementation accurately reproduces fundamental NLSE phenomena including soliton collisions with analytically predicted phase shifts, Akhmediev breather dynamics, and the development of modulation instability from noisy initial conditions. Four canonical test cases validate the numerical scheme: single soliton propagation, two-soliton elastic collision, breather evolution, and noise-seeded modulation instability. The solver employs a 2/3 dealiasing rule with exponential filtering to prevent aliasing errors from the cubic nonlinearity. Statistical analysis using Shannon, R\'enyi, and Tsallis entropies quantifies the spatio-temporal complexity of solutions, while phase space representations reveal the underlying coherence structure. The implementation prioritizes code transparency and educational accessibility over computational performance, providing a valuable pedagogical tool for exploring nonlinear wave dynamics. Complete source code, documentation, and example configurations are freely available, enabling reproducible computational experiments across diverse physical contexts where the NLSE governs wave evolution, including nonlinear optics, Bose-Einstein condensates, and ocean surface waves.

Continuous space-time video super-resolution (C-STVSR) aims to simultaneously enhance video resolution and frame rate at an arbitrary scale. Recently, implicit neural representation (INR) has been applied to video restoration, representing videos as implicit fields that can be decoded at an arbitrary scale. However, the highly ill-posed nature of C-STVSR limits the effectiveness of current INR-based methods: they assume linear motion between frames and use interpolation or feature warping to generate features at arbitrary spatiotemporal positions with two consecutive frames. This restrains C-STVSR from capturing rapid and nonlinear motion and long-term dependencies (involving more than two frames) in complex dynamic scenes. In this paper, we propose a novel C-STVSR framework, called HR-INR, which captures both holistic dependencies and regional motions based on INR. It is assisted by an event camera, a novel sensor renowned for its high temporal resolution and low latency. To fully utilize the rich temporal information from events, we design a feature extraction consisting of (1) a regional event feature extractor - taking events as inputs via the proposed event temporal pyramid representation to capture the regional nonlinear motion and (2) a holistic event-frame feature extractor for long-term dependence and continuity motion. We then propose a novel INR-based decoder with spatiotemporal embeddings to capture long-term dependencies with a larger temporal perception field. We validate the effectiveness and generalization of our method on four datasets (both simulated and real data), showing the superiority of our method.

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of R^n, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. We show that for smooth convex functionals, a W^{2,infty} critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most n-p_0, for some p_0 in (2,3). We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

Tracking by detection has been the prevailing paradigm in the field of Multi-object Tracking (MOT). These methods typically rely on the Kalman Filter to estimate the future locations of objects, assuming linear object motion. However, they fall short when tracking objects exhibiting nonlinear and diverse motion in scenarios like dancing and sports. In addition, there has been limited focus on utilizing learning-based motion predictors in MOT. To address these challenges, we resort to exploring data-driven motion prediction methods. Inspired by the great expectation of state space models (SSMs), such as Mamba, in long-term sequence modeling with near-linear complexity, we introduce a Mamba-based motion model named Mamba moTion Predictor (MTP). MTP is designed to model the complex motion patterns of objects like dancers and athletes. Specifically, MTP takes the spatial-temporal location dynamics of objects as input, captures the motion pattern using a bi-Mamba encoding layer, and predicts the next motion. In real-world scenarios, objects may be missed due to occlusion or motion blur, leading to premature termination of their trajectories. To tackle this challenge, we further expand the application of MTP. We employ it in an autoregressive way to compensate for missing observations by utilizing its own predictions as inputs, thereby contributing to more consistent trajectories. Our proposed tracker, MambaTrack, demonstrates advanced performance on benchmarks such as Dancetrack and SportsMOT, which are characterized by complex motion and severe occlusion.

Creating new visual concepts often requires connecting distinct ideas through their most relevant shared attributes -- their vibe. We introduce Vibe Blending, a novel task for generating coherent and meaningful hybrids that reveals these shared attributes between images. Achieving such blends is challenging for current methods, which struggle to identify and traverse nonlinear paths linking distant concepts in latent space. We propose Vibe Space, a hierarchical graph manifold that learns low-dimensional geodesics in feature spaces like CLIP, enabling smooth and semantically consistent transitions between concepts. To evaluate creative quality, we design a cognitively inspired framework combining human judgments, LLM reasoning, and a geometric path-based difficulty score. We find that Vibe Space produces blends that humans consistently rate as more creative and coherent than current methods.

The multilayer perceptron (MLP), a fundamental paradigm in current artificial intelligence, is widely applied in fields such as computer vision and natural language processing. However, the recently proposed Kolmogorov-Arnold Network (KAN), based on nonlinear additive connections, has been proven to achieve performance comparable to MLPs with significantly fewer parameters. Despite this potential, the use of a single activation function space results in reduced performance of KAN and related works across different tasks. To address this issue, we propose an activation space Selectable KAN (S-KAN). S-KAN employs an adaptive strategy to choose the possible activation mode for data at each feedforward KAN node. Our approach outperforms baseline methods in seven representative function fitting tasks and significantly surpasses MLP methods with the same level of parameters. Furthermore, we extend the structure of S-KAN and propose an activation space selectable Convolutional KAN (S-ConvKAN), which achieves leading results on four general image classification datasets. Our method mitigates the performance variability of the original KAN across different tasks and demonstrates through extensive experiments that feedforward KANs with selectable activations can achieve or even exceed the performance of MLP-based methods. This work contributes to the understanding of the data-centric design of new AI paradigms and provides a foundational reference for innovations in KAN-based network architectures.

This paper introduces Least Volume-a simple yet effective regularization inspired by geometric intuition-that can reduce the necessary number of latent dimensions needed by an autoencoder without requiring any prior knowledge of the intrinsic dimensionality of the dataset. We show that the Lipschitz continuity of the decoder is the key to making it work, provide a proof that PCA is just a linear special case of it, and reveal that it has a similar PCA-like importance ordering effect when applied to nonlinear models. We demonstrate the intuition behind the regularization on some pedagogical toy problems, and its effectiveness on several benchmark problems, including MNIST, CIFAR-10 and CelebA.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

State space models (SSMs) have shown remarkable empirical performance on many long sequence modeling tasks, but a theoretical understanding of these models is still lacking. In this work, we study the learning dynamics of linear SSMs to understand how covariance structure in data, latent state size, and initialization affect the evolution of parameters throughout learning with gradient descent. We show that focusing on the learning dynamics in the frequency domain affords analytical solutions under mild assumptions, and we establish a link between one-dimensional SSMs and the dynamics of deep linear feed-forward networks. Finally, we analyze how latent state over-parameterization affects convergence time and describe future work in extending our results to the study of deep SSMs with nonlinear connections. This work is a step toward a theory of learning dynamics in deep state space models.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

Calculating the action and the interaction Hamiltonian at higher orders in cosmological perturbation theory is a cumbersome task. We employ the formalism of EFT of inflation in models of single field ultra slow-roll inflation and obtain a non-perturbative result for the Hamiltonian in terms of the Goldstone field pi. To complete the dictionary, a non-linear relation between the curvature perturbations and pi is presented. Equipped with these non-linear results, we calculate the higher order loop corrections in USR models which are employed for PBHs formation. It is shown that the loop corrections on long CMB scales increase rapidly with the number of loop L and the setup will go out of perturbative control at the four-loop level.

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory and principal componant analysis can be used when systems undergo low-frequency vibrations. In the case of fast dynamics and wave propagation, we investigate a random generator of boundary conditions for fast submodels by using machine learning. We show that the use of non-linear techniques in machine learning and data-driven methods is highly relevant. Physics-informed neural networks is a possible choice for a data-driven method to replace linear modal analysis. An architecture that support a random component is necessary for the construction of the stochastic model of the physical system for non-parametric uncertainties, since the goal is to learn the underlying probabilistic distribution of uncertainty in the data. Generative Adversarial Networks (GANs) are suited for such applications, where the Wasserstein-GAN with gradient penalty variant offers improved convergence results for our problem. The objective of our approach is to train a GAN on data from a finite element method code (Fenics) so as to extract stochastic boundary conditions for faster finite element predictions on a submodel. The submodel and the training data have both the same geometrical support. It is a zone of interest for uncertainty quantification and relevant to engineering purposes. In the exploitation phase, the framework can be viewed as a randomized and parametrized simulation generator on the submodel, which can be used as a Monte Carlo estimator.

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

The accurate modeling of dynamics in interactive environments is critical for successful long-range prediction. Such a capability could advance Reinforcement Learning (RL) and Planning algorithms, but achieving it is challenging. Inaccuracies in model estimates can compound, resulting in increased errors over long horizons. We approach this problem from the lens of Koopman theory, where the nonlinear dynamics of the environment can be linearized in a high-dimensional latent space. This allows us to efficiently parallelize the sequential problem of long-range prediction using convolution while accounting for the agent's action at every time step. Our approach also enables stability analysis and better control over gradients through time. Taken together, these advantages result in significant improvement over the existing approaches, both in the efficiency and the accuracy of modeling dynamics over extended horizons. We also show that this model can be easily incorporated into dynamics modeling for model-based planning and model-free RL and report promising experimental results.

We present a model for chi^{(2)} waveguides accounting for three modes, two of which make an avoided crossing at the second harmonic wavelength. We introduce two linearly coupled pure modes and adjust the coupling to replicate the waveguide dispersion near the avoided crossing. Analysis of the nonlinear system reveals continuous wave (CW) solutions across much of the parameter-space and prevalence of its modulational instability. We also predict the existence of the avoided-crossing solitons, and study peculiarities of their dynamics and spectral properties, which include formation of a pedestal in the pulse tails and associated pronounced spectral peaks. Mapping these solitons onto the linear dispersion diagrams, we make connections between their existence and CW existence and stability. We also simulate the two-color soliton generation from a single frequency pump pulse to back up its formation and stability properties.

This article investigates signal estimation in wireless transmission (i.e., receive beamforming) from the perspective of statistical machine learning, where the transmit signals may be from an integrated sensing and communication system; that is, 1) signals may be not only discrete constellation points but also arbitrary complex values; 2) signals may be spatially correlated. Particular attention is paid to handling various uncertainties such as the uncertainty of the transmit signal covariance, the uncertainty of the channel matrix, the uncertainty of the channel noise covariance, the existence of channel impulse noises, and the limited sample size of pilots. To proceed, a distributionally robust machine learning framework that is insensitive to the above uncertainties is proposed, which reveals that channel estimation is not a necessary operation. For optimal linear estimation, the proposed framework includes several existing beamformers as special cases such as diagonal loading and eigenvalue thresholding. For optimal nonlinear estimation, estimators are limited in reproducing kernel Hilbert spaces and neural network function spaces, and corresponding uncertainty-aware solutions (e.g., kernelized diagonal loading) are derived. In addition, we prove that the ridge and kernel ridge regression methods in machine learning are distributionally robust against diagonal perturbation in feature covariance.

Generative Adversarial Networks (GANs) have been widely applied in modeling diverse image distributions. However, despite its impressive applications, the structure of the latent space in GANs largely remains as a black-box, leaving its controllable generation an open problem, especially when spurious correlations between different semantic attributes exist in the image distributions. To address this problem, previous methods typically learn linear directions or individual channels that control semantic attributes in the image space. However, they often suffer from imperfect disentanglement, or are unable to obtain multi-directional controls. In this work, in light of the above challenges, we propose a novel approach that discovers nonlinear controls, which enables multi-directional manipulation as well as effective disentanglement, based on gradient information in the learned GAN latent space. More specifically, we first learn interpolation directions by following the gradients from classification networks trained separately on the attributes, and then navigate the latent space by exclusively controlling channels activated for the target attribute in the learned directions. Empirically, with small training data, our approach is able to gain fine-grained controls over a diverse set of bi-directional and multi-directional attributes, and we showcase its ability to achieve disentanglement significantly better than state-of-the-art methods both qualitatively and quantitatively.

Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based solutions face important challenges when dealing with spatio-temporal PDEs over long time scales. Specifically, the current theory of NOs does not present a systematic framework to perform data assimilation and efficiently correct the evolution of PDE solutions over time based on sparsely sampled noisy measurements. In this paper, we propose a learning-based state-space approach to compute the solution operators to infinite-dimensional semilinear PDEs. Exploiting the structure of semilinear PDEs and the theory of nonlinear observers in function spaces, we develop a flexible recursive method that allows for both prediction and data assimilation by combining prediction and correction operations. The proposed framework is capable of producing fast and accurate predictions over long time horizons, dealing with irregularly sampled noisy measurements to correct the solution, and benefits from the decoupling between the spatial and temporal dynamics of this class of PDEs. We show through experiments on the Kuramoto-Sivashinsky, Navier-Stokes and Korteweg-de Vries equations that the proposed model is robust to noise and can leverage arbitrary amounts of measurements to correct its prediction over a long time horizon with little computational overhead.

Koopman representations aim to learn features of nonlinear dynamical systems (NLDS) which lead to linear dynamics in the latent space. Theoretically, such features can be used to simplify many problems in modeling and control of NLDS. In this work we study autoencoder formulations of this problem, and different ways they can be used to model dynamics, specifically for future state prediction over long horizons. We discover several limitations of predicting future states in the latent space and propose an inference-time mechanism, which we refer to as Periodic Reencoding, for faithfully capturing long term dynamics. We justify this method both analytically and empirically via experiments in low and high dimensional NLDS.

Neural vocoders often struggle with aliasing in latent feature spaces, caused by time-domain nonlinear operations and resampling layers. Aliasing folds high-frequency components into the low-frequency range, making aliased and original frequency components indistinguishable and introducing two practical issues. First, aliasing complicates the waveform generation process, as the subsequent layers must address these aliasing effects, increasing the computational complexity. Second, it limits extrapolation performance, particularly in handling high fundamental frequencies, which degrades the perceptual quality of generated speech waveforms. This paper demonstrates that 1) time-domain nonlinear operations inevitably introduce aliasing but provide a strong inductive bias for harmonic generation, and 2) time-frequency-domain processing can achieve aliasing-free waveform synthesis but lacks the inductive bias for effective harmonic generation. Building on this insight, we propose Wavehax, an aliasing-free neural WAVEform generator that integrates 2D convolution and a HArmonic prior for reliable Complex Spectrogram estimation. Experimental results show that Wavehax achieves speech quality comparable to existing high-fidelity neural vocoders and exhibits exceptional robustness in scenarios requiring high fundamental frequency extrapolation, where aliasing effects become typically severe. Moreover, Wavehax requires less than 5% of the multiply-accumulate operations and model parameters compared to HiFi-GAN V1, while achieving over four times faster CPU inference speed.

Real-world time series are often governed by complex nonlinear dynamics. Understanding these underlying dynamics is crucial for precise future prediction. While deep learning has achieved major success in time series forecasting, many existing approaches do not explicitly model the dynamics. To bridge this gap, we introduce DeepEDM, a framework that integrates nonlinear dynamical systems modeling with deep neural networks. Inspired by empirical dynamic modeling (EDM) and rooted in Takens' theorem, DeepEDM presents a novel deep model that learns a latent space from time-delayed embeddings, and employs kernel regression to approximate the underlying dynamics, while leveraging efficient implementation of softmax attention and allowing for accurate prediction of future time steps. To evaluate our method, we conduct comprehensive experiments on synthetic data of nonlinear dynamical systems as well as real-world time series across domains. Our results show that DeepEDM is robust to input noise, and outperforms state-of-the-art methods in forecasting accuracy. Our code is available at: https://abrarmajeedi.github.io/deep_edm.

We demonstrate that violent kinetic preheating following inflation can lead to the formation of black holes in the early Universe. In alpha-attractor models with derivative inflaton couplings, nonlinear amplification of field fluctuations drives large spacetime curvature and gravitational collapse shortly after inflation ends. Using fully general-relativistic lattice simulations, we find that these dynamics produce black holes with masses of order tens of grams at sub-horizon scales, without requiring large primordial curvature perturbations. Although such micro-black holes evaporate rapidly via Hawking radiation, their formation modifies the post-inflationary equation of state and their evaporation can successfully reheat the Universe before Big Bang nucleosynthesis. These results identify kinetic preheating as a new, efficient channel for black-hole production and establish a direct connection between inflationary symmetries and strong-gravity phenomena at reheating.

Diffusion models have recently emerged as powerful generative priors for solving inverse problems. However, training diffusion models in the pixel space are both data-intensive and computationally demanding, which restricts their applicability as priors for high-dimensional real-world data such as medical images. Latent diffusion models, which operate in a much lower-dimensional space, offer a solution to these challenges. However, incorporating latent diffusion models to solve inverse problems remains a challenging problem due to the nonlinearity of the encoder and decoder. To address these issues, we propose ReSample, an algorithm that can solve general inverse problems with pre-trained latent diffusion models. Our algorithm incorporates data consistency by solving an optimization problem during the reverse sampling process, a concept that we term as hard data consistency. Upon solving this optimization problem, we propose a novel resampling scheme to map the measurement-consistent sample back onto the noisy data manifold and theoretically demonstrate its benefits. Lastly, we apply our algorithm to solve a wide range of linear and nonlinear inverse problems in both natural and medical images, demonstrating that our approach outperforms existing state-of-the-art approaches, including those based on pixel-space diffusion models.

While theoretically appealing, the application of the Wasserstein distance to large-scale machine learning problems has been hampered by its prohibitive computational cost. The sliced Wasserstein distance and its variants improve the computational efficiency through the random projection, yet they suffer from low accuracy if the number of projections is not sufficiently large, because the majority of projections result in trivially small values. In this work, we propose a new family of distance metrics, called augmented sliced Wasserstein distances (ASWDs), constructed by first mapping samples to higher-dimensional hypersurfaces parameterized by neural networks. It is derived from a key observation that (random) linear projections of samples residing on these hypersurfaces would translate to much more flexible nonlinear projections in the original sample space, so they can capture complex structures of the data distribution. We show that the hypersurfaces can be optimized by gradient ascent efficiently. We provide the condition under which the ASWD is a valid metric and show that this can be obtained by an injective neural network architecture. Numerical results demonstrate that the ASWD significantly outperforms other Wasserstein variants for both synthetic and real-world problems.

Large Reasoning Models (LRMs) have significantly advanced beyond traditional Large Language Models (LLMs) with their exceptional logical reasoning capabilities, yet these improvements introduce heightened safety risks. When subjected to jailbreak attacks, their ability to generate more targeted and organized content can lead to greater harm. Although some studies claim that reasoning enables safer LRMs against existing LLM attacks, they overlook the inherent flaws within the reasoning process itself. To address this gap, we propose the first jailbreak attack targeting LRMs, exploiting their unique vulnerabilities stemming from the advanced reasoning capabilities. Specifically, we introduce a Chaos Machine, a novel component to transform attack prompts with diverse one-to-one mappings. The chaos mappings iteratively generated by the machine are embedded into the reasoning chain, which strengthens the variability and complexity and also promotes a more robust attack. Based on this, we construct the Mousetrap framework, which makes attacks projected into nonlinear-like low sample spaces with mismatched generalization enhanced. Also, due to the more competing objectives, LRMs gradually maintain the inertia of unpredictable iterative reasoning and fall into our trap. Success rates of the Mousetrap attacking o1-mini, Claude-Sonnet and Gemini-Thinking are as high as 96%, 86% and 98% respectively on our toxic dataset Trotter. On benchmarks such as AdvBench, StrongREJECT, and HarmBench, attacking Claude-Sonnet, well-known for its safety, Mousetrap can astonishingly achieve success rates of 87.5%, 86.58% and 93.13% respectively. Attention: This paper contains inappropriate, offensive and harmful content.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

Hyperbolic neural networks have shown great potential for modeling complex data. However, existing hyperbolic networks are not completely hyperbolic, as they encode features in a hyperbolic space yet formalize most of their operations in the tangent space (a Euclidean subspace) at the origin of the hyperbolic space. This hybrid method greatly limits the modeling ability of networks. In this paper, we propose a fully hyperbolic framework to build hyperbolic networks based on the Lorentz model by adapting the Lorentz transformations (including boost and rotation) to formalize essential operations of neural networks. Moreover, we also prove that linear transformation in tangent spaces used by existing hyperbolic networks is a relaxation of the Lorentz rotation and does not include the boost, implicitly limiting the capabilities of existing hyperbolic networks. The experimental results on four NLP tasks show that our method has better performance for building both shallow and deep networks. Our code will be released to facilitate follow-up research.

Cameras currently allow access to two image states: (i) a minimally processed linear raw-RGB image state (i.e., raw sensor data) or (ii) a highly-processed nonlinear image state (e.g., sRGB). There are many computer vision tasks that work best with a linear image state, such as image deblurring and image dehazing. Unfortunately, the vast majority of images are saved in the nonlinear image state. Because of this, a number of methods have been proposed to "unprocess" nonlinear images back to a raw-RGB state. However, existing unprocessing methods have a drawback because raw-RGB images are sensor-specific. As a result, it is necessary to know which camera produced the sRGB output and use a method or network tailored for that sensor to properly unprocess it. This paper addresses this limitation by exploiting another camera image state that is not available as an output, but it is available inside the camera pipeline. In particular, cameras apply a colorimetric conversion step to convert the raw-RGB image to a device-independent space based on the CIE XYZ color space before they apply the nonlinear photo-finishing. Leveraging this canonical image state, we propose a deep learning framework, CIE XYZ Net, that can unprocess a nonlinear image back to the canonical CIE XYZ image. This image can then be processed by any low-level computer vision operator and re-rendered back to the nonlinear image. We demonstrate the usefulness of the CIE XYZ Net on several low-level vision tasks and show significant gains that can be obtained by this processing framework. Code and dataset are publicly available at https://github.com/mahmoudnafifi/CIE_XYZ_NET.

Motivated by the hypothesis that neural network representations encode abstract, interpretable features as linearly accessible, approximately orthogonal directions, sparse autoencoders (SAEs) have become a popular tool in interpretability. However, recent work has demonstrated phenomenology of model representations that lies outside the scope of this hypothesis, showing signatures of hierarchical, nonlinear, and multi-dimensional features. This raises the question: do SAEs represent features that possess structure at odds with their motivating hypothesis? If not, does avoiding this mismatch help identify said features and gain further insights into neural network representations? To answer these questions, we take a construction-based approach and re-contextualize the popular matching pursuits (MP) algorithm from sparse coding to design MP-SAE -- an SAE that unrolls its encoder into a sequence of residual-guided steps, allowing it to capture hierarchical and nonlinearly accessible features. Comparing this architecture with existing SAEs on a mixture of synthetic and natural data settings, we show: (i) hierarchical concepts induce conditionally orthogonal features, which existing SAEs are unable to faithfully capture, and (ii) the nonlinear encoding step of MP-SAE recovers highly meaningful features, helping us unravel shared structure in the seemingly dichotomous representation spaces of different modalities in a vision-language model, hence demonstrating the assumption that useful features are solely linearly accessible is insufficient. We also show that the sequential encoder principle of MP-SAE affords an additional benefit of adaptive sparsity at inference time, which may be of independent interest. Overall, we argue our results provide credence to the idea that interpretability should begin with the phenomenology of representations, with methods emerging from assumptions that fit it.

Instrumental variable methods have been widely used to identify causal effects in the presence of unmeasured confounding. A key identification condition known as the exclusion restriction states that the instrument cannot have a direct effect on the outcome which is not mediated by the exposure in view. In the health and social sciences, such an assumption is often not credible. To address this concern, we consider identification conditions of the population average treatment effect with an invalid instrumental variable which does not satisfy the exclusion restriction, and derive the efficient influence function targeting the identifying functional under a nonparametric observed data model. We propose a novel multiply robust locally efficient estimator of the average treatment effect that is consistent in the union of multiple parametric nuisance models, as well as a multiply debiased machine learning estimator for which the nuisance parameters are estimated using generic machine learning methods, that effectively exploit various forms of linear or nonlinear structured sparsity in the nuisance parameter space. When one cannot be confident that any of these machine learners is consistent at sufficiently fast rates to ensure n-consistency for the average treatment effect, we introduce a new criteria for selective machine learning which leverages the multiple robustness property in order to ensure small bias. The proposed methods are illustrated through extensive simulations and a data analysis evaluating the causal effect of 401(k) participation on savings.

Unsupervised learning of probabilistic models is a central yet challenging problem in machine learning. Specifically, designing models with tractable learning, sampling, inference and evaluation is crucial in solving this task. We extend the space of such models using real-valued non-volume preserving (real NVP) transformations, a set of powerful invertible and learnable transformations, resulting in an unsupervised learning algorithm with exact log-likelihood computation, exact sampling, exact inference of latent variables, and an interpretable latent space. We demonstrate its ability to model natural images on four datasets through sampling, log-likelihood evaluation and latent variable manipulations.

Large language models (LLMs) have achieved remarkable success and demonstrated superior performance across various tasks, including natural language processing (NLP), weather forecasting, biological protein folding, text generation, and solving mathematical problems. However, many real-world data exhibit highly non-Euclidean latent hierarchical anatomy, such as protein networks, transportation networks, financial networks, brain networks, and linguistic structures or syntactic trees in natural languages. Effectively learning intrinsic semantic entailment and hierarchical relationships from these raw, unstructured input data using LLMs remains an underexplored area. Due to its effectiveness in modeling tree-like hierarchical structures, hyperbolic geometry -- a non-Euclidean space -- has rapidly gained popularity as an expressive latent representation space for complex data modeling across domains such as graphs, images, languages, and multi-modal data. Here, we provide a comprehensive and contextual exposition of recent advancements in LLMs that leverage hyperbolic geometry as a representation space to enhance semantic representation learning and multi-scale reasoning. Specifically, the paper presents a taxonomy of the principal techniques of Hyperbolic LLMs (HypLLMs) in terms of four main categories: (1) hyperbolic LLMs through exp/log maps; (2) hyperbolic fine-tuned models; (3) fully hyperbolic LLMs, and (4) hyperbolic state-space models. We also explore crucial potential applications and outline future research directions. A repository of key papers, models, datasets, and code implementations is available at https://github.com/sarangp2402/Hyperbolic-LLM-Models/tree/main.

We prove that: I. If L is a T_1 space, |L|>1 and d(L) leq kappa geq omega, then there is a submaximal dense subspace X of L^{2^kappa} such that |X|=Delta(X)=kappa; II. If cleqkappa=kappa^omega<lambda and 2^kappa=2^lambda, then there is a Tychonoff pseudocompact globally and locally connected space X such that |X|=Delta(X)=lambda and X is not kappa^+-resolvable; III. If omega_1leqkappa<lambda and 2^kappa=2^lambda, then there is a regular space X such that |X|=Delta(X)=lambda, all continuous real-valued functions on X are constant (so X is pseudocompact and connected) and X is not kappa^+-resolvable.

Convolutional networks are large linear systems divided into layers and connected by non-linear units. These units are the "articulations" that allow the network to adapt to the input. To understand how a network manages to solve a problem we must look at the articulated decisions in entirety. If we could capture the actions of non-linear units for a particular input, we would be able to replay the whole system back and forth as if it was always linear. It would also reveal the actions of non-linearities because the resulting linear system, a Linear Interpreter, depends on the input image. We introduce a hooking layer, called a LinearScope, which allows us to run the network and the linear interpreter in parallel. Its implementation is simple, flexible and efficient. From here we can make many curious inquiries: how do these linear systems look like? When the rows and columns of the transformation matrix are images, how do they look like? What type of basis do these linear transformations rely on? The answers depend on the problems presented, through which we take a tour to some popular architectures used for classification, super-resolution (SR) and image-to-image translation (I2I). For classification we observe that popular networks use a pixel-wise vote per class strategy and heavily rely on bias parameters. For SR and I2I we find that CNNs use wavelet-type basis similar to the human visual system. For I2I we reveal copy-move and template-creation strategies to generate outputs.

Adapting standard methods from geometric measure theory, we provide an example of a polynomial-valued measure mu on tame sets in R^d which satisfies many desirable properties. Among these is strict monotonicity: the measure of a proper subset is strictly less than the measure of the whole set. Using techniques from non-standard analysis, we display that the domain of mu can be extended to all subsets of R^d (up to equivalence modulo infinitesimals). The resulting extension is a numerosity function that encodes the i-dimensional Hausdorff measure for all iin N, as well as the i-th intrinsic volume functions.

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

We investigate the construction of latent spaces through self-supervised learning to support semantically meaningful operations. Analogous to operational amplifiers, these "operational latent spaces" (OpLaS) not only demonstrate semantic structure such as clustering but also support common transformational operations with inherent semantic meaning. Some operational latent spaces are found to have arisen "unintentionally" in the progress toward some (other) self-supervised learning objective, in which unintended but still useful properties are discovered among the relationships of points in the space. Other spaces may be constructed "intentionally" by developers stipulating certain kinds of clustering or transformations intended to produce the desired structure. We focus on the intentional creation of operational latent spaces via self-supervised learning, including the introduction of rotation operators via a novel "FiLMR" layer, which can be used to enable ring-like symmetries found in some musical constructions.

It is often useful to compactly summarize important properties of model parameters and training data so that they can be used later without storing and/or iterating over the entire dataset. As a specific case, we consider estimating the Function Space Distance (FSD) over a training set, i.e. the average discrepancy between the outputs of two neural networks. We propose a Linearized Activation Function TRick (LAFTR) and derive an efficient approximation to FSD for ReLU neural networks. The key idea is to approximate the architecture as a linear network with stochastic gating. Despite requiring only one parameter per unit of the network, our approach outcompetes other parametric approximations with larger memory requirements. Applied to continual learning, our parametric approximation is competitive with state-of-the-art nonparametric approximations, which require storing many training examples. Furthermore, we show its efficacy in estimating influence functions accurately and detecting mislabeled examples without expensive iterations over the entire dataset.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square summability of Dirichlet coefficients for every s > 1, with a Banach component controlling logarithmic averages of partial sums. We prove that U is a complete Banach space which embeds continuously all standard Hilbert spaces of Dirichlet series and allows natural actions of Dirichlet convolution and shift operators. This framework provides a unified analytic setting for classical and modern problems in multiplicative number theory.

We present Semantify: a self-supervised method that utilizes the semantic power of CLIP language-vision foundation model to simplify the control of 3D morphable models. Given a parametric model, training data is created by randomly sampling the model's parameters, creating various shapes and rendering them. The similarity between the output images and a set of word descriptors is calculated in CLIP's latent space. Our key idea is first to choose a small set of semantically meaningful and disentangled descriptors that characterize the 3DMM, and then learn a non-linear mapping from scores across this set to the parametric coefficients of the given 3DMM. The non-linear mapping is defined by training a neural network without a human-in-the-loop. We present results on numerous 3DMMs: body shape models, face shape and expression models, as well as animal shapes. We demonstrate how our method defines a simple slider interface for intuitive modeling, and show how the mapping can be used to instantly fit a 3D parametric body shape to in-the-wild images.

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).

We present a Non-parametric Network for 3D point cloud analysis, Point-NN, which consists of purely non-learnable components: farthest point sampling (FPS), k-nearest neighbors (k-NN), and pooling operations, with trigonometric functions. Surprisingly, it performs well on various 3D tasks, requiring no parameters or training, and even surpasses existing fully trained models. Starting from this basic non-parametric model, we propose two extensions. First, Point-NN can serve as a base architectural framework to construct Parametric Networks by simply inserting linear layers on top. Given the superior non-parametric foundation, the derived Point-PN exhibits a high performance-efficiency trade-off with only a few learnable parameters. Second, Point-NN can be regarded as a plug-and-play module for the already trained 3D models during inference. Point-NN captures the complementary geometric knowledge and enhances existing methods for different 3D benchmarks without re-training. We hope our work may cast a light on the community for understanding 3D point clouds with non-parametric methods. Code is available at https://github.com/ZrrSkywalker/Point-NN.

Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is quasar-convexity, a non-convex generalization of convexity that subsumes convex functions. Existing algorithms for minimizing quasar-convex functions in the stochastic setting have either high complexity or slow convergence, which prompts us to derive a new class of stochastic methods for optimizing smooth quasar-convex functions. We demonstrate that our algorithms have fast convergence and outperform existing algorithms on several examples, including the classical problem of learning linear dynamical systems. We also present a unified analysis of our newly proposed algorithms and a previously studied deterministic algorithm.

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and consistently deliver state-of-the-art results across diverse applications. However, hyperbolic classifiers often grapple with computational challenges. Methods reliant on Riemannian optimization frequently exhibit sluggishness, stemming from the increased computational demands of operations on Riemannian manifolds. In response to these challenges, we present hyperDT, a novel extension of decision tree algorithms into hyperbolic space. Crucially, hyperDT eliminates the need for computationally intensive Riemannian optimization, numerically unstable exponential and logarithmic maps, or pairwise comparisons between points by leveraging inner products to adapt Euclidean decision tree algorithms to hyperbolic space. Our approach is conceptually straightforward and maintains constant-time decision complexity while mitigating the scalability issues inherent in high-dimensional Euclidean spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest model. Extensive benchmarking across diverse datasets underscores the superior performance of these models, providing a swift, precise, accurate, and user-friendly toolkit for hyperbolic data analysis.

Neural networks have proven to be a highly effective tool for solving complex problems in many areas of life. Recently, their importance and practical usability have further been reinforced with the advent of deep learning. One of the important conditions for the success of neural networks is the choice of an appropriate activation function introducing non-linearity into the model. Many types of these functions have been proposed in the literature in the past, but there is no single comprehensive source containing their exhaustive overview. The absence of this overview, even in our experience, leads to redundancy and the unintentional rediscovery of already existing activation functions. To bridge this gap, our paper presents an extensive survey involving 400 activation functions, which is several times larger in scale than previous surveys. Our comprehensive compilation also references these surveys; however, its main goal is to provide the most comprehensive overview and systematization of previously published activation functions with links to their original sources. The secondary aim is to update the current understanding of this family of functions.

This paper develops a hierarchical clustering algorithm for weighted projective spaces P_{q}, utilizing a Finsler metric d_F([z], [w]) and its rational analogue d_{F,Q}([z], [w]) to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades q = (q_0, q_1, dots, q_n), these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

In many practical applications of machine learning data arrives sequentially over time in large chunks. Practitioners have then to decide how to allocate their computational budget in order to obtain the best performance at any point in time. Online learning theory for convex optimization suggests that the best strategy is to use data as soon as it arrives. However, this might not be the best strategy when using deep non-linear networks, particularly when these perform multiple passes over each chunk of data rendering the overall distribution non i.i.d.. In this paper, we formalize this learning setting in the simplest scenario in which each data chunk is drawn from the same underlying distribution, and make a first attempt at empirically answering the following questions: How long should the learner wait before training on the newly arrived chunks? What architecture should the learner adopt? Should the learner increase capacity over time as more data is observed? We probe this learning setting using convolutional neural networks trained on classic computer vision benchmarks as well as a large transformer model trained on a large-scale language modeling task. Code is available at www.github.com/facebookresearch/ALMA.

We present Manify, an open-source Python library for non-Euclidean representation learning. Leveraging manifold learning techniques, Manify provides tools for learning embeddings in (products of) non-Euclidean spaces, performing classification and regression with data that lives in such spaces, and estimating the curvature of a manifold. Manify aims to advance research and applications in machine learning by offering a comprehensive suite of tools for manifold-based data analysis. Our source code, examples, datasets, results, and documentation are available at https://github.com/pchlenski/manify

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. the NTK parametrization). However, we show that the standard and NTK parametrizations of a neural network do not admit infinite-width limits that can learn features, which is crucial for pretraining and transfer learning such as with BERT. We propose simple modifications to the standard parametrization to allow for feature learning in the limit. Using the *Tensor Programs* technique, we derive explicit formulas for such limits. On Word2Vec and few-shot learning on Omniglot via MAML, two canonical tasks that rely crucially on feature learning, we compute these limits exactly. We find that they outperform both NTK baselines and finite-width networks, with the latter approaching the infinite-width feature learning performance as width increases. More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. We show 1) any parametrization in this space either admits feature learning or has an infinite-width training dynamics given by kernel gradient descent, but not both; 2) any such infinite-width limit can be computed using the Tensor Programs technique. Code for our experiments can be found at github.com/edwardjhu/TP4.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function f_theta (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function f_theta follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

The majority of quantization methods have been proposed to reduce the model size of Vision Transformers, yet most of them have overlooked the quantization of non-linear operations. Only a few works have addressed quantization for non-linear operations, but they applied a single quantization method across all non-linear operations. We believe that this can be further improved by employing a different quantization method for each non-linear operation. Therefore, to assign the most error-minimizing quantization method from the known methods to each non-linear layer, we propose a mixed non-linear quantization that considers layer-wise quantization sensitivity measured by SQNR difference metric. The results show that our method outperforms I-BERT, FQ-ViT, and I-ViT in both 8-bit and 6-bit settings for ViT, DeiT, and Swin models by an average of 0.6%p and 19.6%p, respectively. Our method outperforms I-BERT and I-ViT by 0.6%p and 20.8%p, respectively, when training time is limited. We plan to release our code at https://gitlab.com/ones-ai/mixed-non-linear-quantization.

Modeling the human body in a canonical space is a common practice for capturing and animation. But when involving the neural radiance field (NeRF), learning a static NeRF in the canonical space is not enough because the lighting of the body changes when the person moves even though the scene lighting is constant. Previous methods alleviate the inconsistency of lighting by learning a per-frame embedding, but this operation does not generalize to unseen poses. Given that the lighting condition is static in the world space while the human body is consistent in the canonical space, we propose a dual-space NeRF that models the scene lighting and the human body with two MLPs in two separate spaces. To bridge these two spaces, previous methods mostly rely on the linear blend skinning (LBS) algorithm. However, the blending weights for LBS of a dynamic neural field are intractable and thus are usually memorized with another MLP, which does not generalize to novel poses. Although it is possible to borrow the blending weights of a parametric mesh such as SMPL, the interpolation operation introduces more artifacts. In this paper, we propose to use the barycentric mapping, which can directly generalize to unseen poses and surprisingly achieves superior results than LBS with neural blending weights. Quantitative and qualitative results on the Human3.6M and the ZJU-MoCap datasets show the effectiveness of our method.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

In this work we introduce Steerable Transformers, an extension of the Vision Transformer mechanism that maintains equivariance to the special Euclidean group SE(d). We propose an equivariant attention mechanism that operates on features extracted by steerable convolutions. Operating in Fourier space, our network utilizes Fourier space non-linearities. Our experiments in both two and three dimensions show that adding a steerable transformer encoder layer to a steerable convolution network enhances performance.

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regressionx2014which generalizes standard regression to metric-space valued datax2014the method estimates a mean path on either side of a candidate cutoff, extending existing k-sample tests to a flexible regression setting. Key theoretical contributions include a central limit theorem for the local estimator of the conditional Fr\'echet variance and the asymptotic validity and consistency of the proposed test. Simulations confirm nominal size control and robust power in finite samples. Two applications demonstrate the method's value by revealing effects invisible to scalar-based tests. First, I detect a sharp change in work-from-home compositions at Washington State's income threshold for non-compete enforceability during COVID-19, highlighting remote work's role as a bargaining margin. Second, I find that countries restructure their input-output networks after losing preferential US trade access. These findings underscore that analyzing regression functions within their native metric spaces can reveal structural discontinuities that scalar summaries would miss.

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.