Daily Papers

We introduce ROSE, a deep-learning exchange--correlation energy functional built from the occupied-state density matrix. On GMTKN55 and on additional external benchmarks considered in this work, ROSE improves over modern neural network functionals trained under matched protocols and is on average more accurate than, or on par with, most hybrid and range-separated hybrid functionals considered here. On the experimental benchmarks considered here it has leading accuracy across hybrids, range separated hybrids, and neural functionals. These results motivate nonlocal density-matrix representations as a route to systematic improvement of learned XC functionals with low computational cost.

Both convolutional and recurrent operations are building blocks that process one local neighborhood at a time. In this paper, we present non-local operations as a generic family of building blocks for capturing long-range dependencies. Inspired by the classical non-local means method in computer vision, our non-local operation computes the response at a position as a weighted sum of the features at all positions. This building block can be plugged into many computer vision architectures. On the task of video classification, even without any bells and whistles, our non-local models can compete or outperform current competition winners on both Kinetics and Charades datasets. In static image recognition, our non-local models improve object detection/segmentation and pose estimation on the COCO suite of tasks. Code is available at https://github.com/facebookresearch/video-nonlocal-net .

Large language models (LLMs) can be adapted to new tasks using parameter-efficient fine-tuning (PEFT) methods that modify only a small number of trainable parameters, often through low-rank updates. In this work, we adopt a quantum-information-inspired perspective to understand their effectiveness. From this perspective, low-rank parameterizations naturally correspond to low-dimensional Matrix Product States (MPS) representations, which enable entanglement-based characterizations of parameter structure. Thereby, we term and measure "Artificial Entanglement", defined as the entanglement entropy of the parameters in artificial neural networks (in particular the LLMs). We first study the representative low-rank adaptation (LoRA) PEFT method, alongside full fine-tuning (FFT), using LLaMA models at the 1B and 8B scales trained on the Tulu3 and OpenThoughts3 datasets, and uncover: (i) Internal artificial entanglement in the updates of query and value projection matrices in LoRA follows a volume law with a central suppression (termed as the "Entanglement Valley"), which is sensitive to hyper-parameters and is distinct from that in FFT; (ii) External artificial entanglement in attention matrices, corresponding to token-token correlations in representation space, follows an area law with logarithmic corrections and remains robust to LoRA hyper-parameters and training steps. Drawing a parallel to the No-Hair Theorem in black hole physics, we propose that although LoRA and FFT induce distinct internal entanglement signatures, such differences do not manifest in the attention outputs, suggesting a "no-hair" property that results in the effectiveness of low rank updates. We further provide theoretical support based on random matrix theory, and extend our analysis to an MPS Adaptation PEFT method, which exhibits qualitatively similar behaviors.

Models of physical phenomena that use nonlocal operators are better suited for some applications than their classical counterparts that employ partial differential operators. However, the numerical solution of these nonlocal problems can be quite expensive. Therefore, Local-to-Nonlocal couplings have emerged that combine partial differential operators with nonlocal operators. In this work, we make use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem.

Neural architectures that learn potential energy surfaces from molecular data have undergone fast improvement in recent years. A key driver of this success is the Message Passing Neural Network (MPNN) paradigm. Its favorable scaling with system size partly relies upon a spatial distance limit on messages. While this focus on locality is a useful inductive bias, it also impedes the learning of long-range interactions such as electrostatics and van der Waals forces. To address this drawback, we propose Ewald message passing: a nonlocal Fourier space scheme which limits interactions via a cutoff on frequency instead of distance, and is theoretically well-founded in the Ewald summation method. It can serve as an augmentation on top of existing MPNN architectures as it is computationally inexpensive and agnostic to architectural details. We test the approach with four baseline models and two datasets containing diverse periodic (OC20) and aperiodic structures (OE62). We observe robust improvements in energy mean absolute errors across all models and datasets, averaging 10% on OC20 and 16% on OE62. Our analysis shows an outsize impact of these improvements on structures with high long-range contributions to the ground truth energy.

We introduce a family of random matrices where correlations between matrix elements are induced via interaction-derived Boltzmann factors. Varying these yields access to different ensembles. We find a universal scaling behavior of the finite-size statistics characterized by a heavy-tailed eigenvalue distribution whose extremes are governed by the Fréchet extreme value distribution for the case corresponding to a ferromagnetic Ising transition. The introduction of a finite density of nonlocal interactions restores standard random-matrix behavior. Suitably rescaled average extremes, playing a physical role as an order parameter, can thus discriminate aspects of the interaction structure; they also yield further nonuniversal information. In particular, the link between maximum eigenvalues and order parameters offers a potential route to resolving long-standing problems in statistical physics, such as deriving the exact magnetization scaling function in the two-dimensional Ising model at criticality.

Nonlinear operators with long distance spatiotemporal dependencies are fundamental in modeling complex systems across sciences, yet learning these nonlocal operators remains challenging in machine learning. Integral equations (IEs), which model such nonlocal systems, have wide ranging applications in physics, chemistry, biology, and engineering. We introduce Neural Integral Equations (NIE), a method for learning unknown integral operators from data using an IE solver. To improve scalability and model capacity, we also present Attentional Neural Integral Equations (ANIE), which replaces the integral with self-attention. Both models are grounded in the theory of second kind integral equations, where the indeterminate appears both inside and outside the integral operator. We provide theoretical analysis showing how self-attention can approximate integral operators under mild regularity assumptions, further deepening previously reported connections between transformers and integration, and deriving corresponding approximation results for integral operators. Through numerical benchmarks on synthetic and real world data, including Lotka-Volterra, Navier-Stokes, and Burgers' equations, as well as brain dynamics and integral equations, we showcase the models' capabilities and their ability to derive interpretable dynamics embeddings. Our experiments demonstrate that ANIE outperforms existing methods, especially for longer time intervals and higher dimensional problems. Our work addresses a critical gap in machine learning for nonlocal operators and offers a powerful tool for studying unknown complex systems with long range dependencies.

Recently, sophisticated deep learning-based approaches have been developed for generating efficient initial guesses to accelerate the convergence of density functional theory (DFT) calculations. While the actual initial guesses are often density matrices (DM), quantities that can convert into density matrices also qualify as alternative forms of initial guesses. Hence, existing works mostly rely on the prediction of the Hamiltonian matrix for obtaining high-quality initial guesses. However, the Hamiltonian matrix is both numerically difficult to predict and intrinsically non-transferable, hindering the application of such models in real scenarios. In light of this, we propose a method that constructs DFT initial guesses by predicting the electron density in a compact auxiliary basis representation using E(3)-equivariant neural networks. Trained on small molecules with up to 20 atoms, our model is able to achieve an average 33.3% self-consistent field (SCF) step reduction on systems up to 60 atoms, substantially outperforming Hamiltonian-centric and DM-centric models. Critically, this acceleration remains nearly constant with increasing system sizes and exhibits strong transferring behaviors across orbital basis sets and exchange-correlation (XC) functionals. To the best of our knowledge, this work represents the first and robust candidate for a universally transferable DFT acceleration method. We are also releasing the SCFbench dataset and its accompanying code to facilitate future research in this promising direction.

Recently, Neural Radiance Field (NeRF) has shown great success in rendering novel-view images of a given scene by learning an implicit representation with only posed RGB images. NeRF and relevant neural field methods (e.g., neural surface representation) typically optimize a point-wise loss and make point-wise predictions, where one data point corresponds to one pixel. Unfortunately, this line of research failed to use the collective supervision of distant pixels, although it is known that pixels in an image or scene can provide rich structural information. To the best of our knowledge, we are the first to design a nonlocal multiplex training paradigm for NeRF and relevant neural field methods via a novel Stochastic Structural SIMilarity (S3IM) loss that processes multiple data points as a whole set instead of process multiple inputs independently. Our extensive experiments demonstrate the unreasonable effectiveness of S3IM in improving NeRF and neural surface representation for nearly free. The improvements of quality metrics can be particularly significant for those relatively difficult tasks: e.g., the test MSE loss unexpectedly drops by more than 90% for TensoRF and DVGO over eight novel view synthesis tasks; a 198% F-score gain and a 64% Chamfer L_{1} distance reduction for NeuS over eight surface reconstruction tasks. Moreover, S3IM is consistently robust even with sparse inputs, corrupted images, and dynamic scenes.

Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.

The Schr\"odinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space R^{D} and leaves open theoretical and algorithmic questions about applying SB methods to discrete data, e.g, on finite spaces S^{D}. Notable examples of such sets S are codebooks of vector-quantized (VQ) representations of modern autoencoders, tokens in texts, categories of atoms in molecules, etc. In this paper, we provide a theoretical and algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. Specifically, we theoretically justify the convergence of discrete-time IMF (D-IMF) to SB in discrete spaces. This enables us to develop a practical computational algorithm for SB which we call Categorical Schr\"odinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images.

Machine learning (ML) holds great promise for extracting insights from complex quantum many-body data obtained in quantum experiments. This approach can efficiently solve certain quantum problems that are classically intractable, suggesting potential advantages of harnessing quantum data. However, addressing large-scale problems still requires significant amounts of data beyond the limited computational resources of near-term quantum devices. We propose a scalable ML framework called Geometrically Local Quantum Kernel (GLQK), designed to efficiently learn quantum many-body experimental data by leveraging the exponential decay of correlations, a phenomenon prevalent in noncritical systems. In the task of learning an unknown polynomial of quantum expectation values, we rigorously prove that GLQK substantially improves polynomial sample complexity in the number of qubits n, compared to the existing shadow kernel, by constructing a feature space from local quantum information at the correlation length scale. This improvement is particularly notable when each term of the target polynomial involves few local subsystems. Remarkably, for translationally symmetric data, GLQK achieves constant sample complexity, independent of n. We numerically demonstrate its high scalability in two learning tasks on quantum many-body phenomena. These results establish new avenues for utilizing experimental data to advance the understanding of quantum many-body physics.

Density ratio estimation is fundamental to tasks involving f-divergences, yet existing methods often fail under significantly different distributions or inadequately overlapping supports -- the density-chasm and the support-chasm problems. Additionally, prior approaches yield divergent time scores near boundaries, leading to instability. We design D^3RE, a unified framework for robust, stable and efficient density ratio estimation. We propose the dequantified diffusion bridge interpolant (DDBI), which expands support coverage and stabilizes time scores via diffusion bridges and Gaussian dequantization. Building on DDBI, the proposed dequantified Schr{\"o}dinger bridge interpolant (DSBI) incorporates optimal transport to solve the Schr{\"o}dinger bridge problem, enhancing accuracy and efficiency. Our method offers uniform approximation and bounded time scores in theory, and outperforms baselines empirically in mutual information and density estimation tasks.

We describe nonparametric deconvolution models (NDMs), a family of Bayesian nonparametric models for collections of data in which each observation is the average over the features from heterogeneous particles. For example, these types of data are found in elections, where we observe precinct-level vote tallies (observations) of individual citizens' votes (particles) across each of the candidates or ballot measures (features), where each voter is part of a specific voter cohort or demographic (factor). Like the hierarchical Dirichlet process, NDMs rely on two tiers of Dirichlet processes to explain the data with an unknown number of latent factors; each observation is modeled as a weighted average of these latent factors. Unlike existing models, NDMs recover how factor distributions vary locally for each observation. This uniquely allows NDMs both to deconvolve each observation into its constituent factors, and also to describe how the factor distributions specific to each observation vary across observations and deviate from the corresponding global factors. We present variational inference techniques for this family of models and study its performance on simulated data and voting data from California. We show that including local factors improves estimates of global factors and provides a novel scaffold for exploring data.

The successes of modern deep machine learning methods are founded on their ability to transform inputs across multiple layers to build good high-level representations. It is therefore critical to understand this process of representation learning. However, standard theoretical approaches (formally NNGPs) involving infinite width limits eliminate representation learning. We therefore develop a new infinite width limit, the Bayesian representation learning limit, that exhibits representation learning mirroring that in finite-width models, yet at the same time, retains some of the simplicity of standard infinite-width limits. In particular, we show that Deep Gaussian processes (DGPs) in the Bayesian representation learning limit have exactly multivariate Gaussian posteriors, and the posterior covariances can be obtained by optimizing an interpretable objective combining a log-likelihood to improve performance with a series of KL-divergences which keep the posteriors close to the prior. We confirm these results experimentally in wide but finite DGPs. Next, we introduce the possibility of using this limit and objective as a flexible, deep generalisation of kernel methods, that we call deep kernel machines (DKMs). Like most naive kernel methods, DKMs scale cubically in the number of datapoints. We therefore use methods from the Gaussian process inducing point literature to develop a sparse DKM that scales linearly in the number of datapoints. Finally, we extend these approaches to NNs (which have non-Gaussian posteriors) in the Appendices.

Dense associative memories (DAMs) store and retrieve patterns via energy-function based fixed points, but existing models are limited to vector representations. We extend DAMs to Gaussian densities equipped with the 2-Wasserstein distance. Our framework defines a log-sum-exp energy over stored distributions and a retrieval dynamics aggregating optimal transport maps in a Gibbs-weighted manner. Stationary points correspond to self-consistent Wasserstein barycenters, generalizing classical DAM fixed points. We prove exponential storage capacity and provide quantitative retrieval guarantees under Wasserstein perturbations. We validate the method on synthetic and real-world image (CelebA and CIFAR-10 datasets) and text (text8 and NLI corpus) datasets. By generalizing from vectors to distributions, our work bridges classical DAMs with modern generative modeling and paves way for distributional storage and retrieval in memory-augmented learning.

We propose a hierarchical attention mechanism based on two-level overlapping Schwarz domain decomposition. The method is motivated by the observation that two-level Schwarz domain decomposition methods combine local subdomain corrections with a coarse level that communicates global, long-range information. We test its usefulness in the context of finite-dimensional operator learning using a simple, one-dimensional diffusion problem with homogeneous Dirichlet boundary conditions. Although elementary, this problem provides a controlled sequence-to-sequence setting in which the exact nonlocal solution operator is known. After discretization, learning the solution operator amounts to approximating the inverse of a symmetric positive definite matrix. As a baseline, we use a global softmax-free low-rank attention operator of the form QK^T. The proposed construction replaces this dense global factorization by a two-level additive structure: local low-rank attention blocks on overlapping subdomains are combined with a coarse attention block. The resulting operator has the form $M_θ^{-1} = ΦQ_0 K_0^T Φ^T + sum_{i=1}^{N} R_i^T D_i^{1/2} Q_i K_i^T D_i^{1/2} R_i. Here R_i restricts to an overlapping subdomain, D_i is a partition-of-unity weight, and Φ$ is a coarse interpolation (or prolongation) matrix. Numerical experiments for synthetic Fourier right-hand sides indicate that the domain-decomposition attention operator is able to train faster and can give more accurate approximations than a global low-rank attention baseline while using significantly fewer parameters.

Uncertainty Quantification (UQ) is essential for creating trustworthy machine learning models. Recent years have seen a steep rise in UQ methods that can flag suspicious examples, however, it is often unclear what exactly these methods identify. In this work, we propose a framework for categorizing uncertain examples flagged by UQ methods in classification tasks. We introduce the confusion density matrix -- a kernel-based approximation of the misclassification density -- and use this to categorize suspicious examples identified by a given uncertainty method into three classes: out-of-distribution (OOD) examples, boundary (Bnd) examples, and examples in regions of high in-distribution misclassification (IDM). Through extensive experiments, we show that our framework provides a new and distinct perspective for assessing differences between uncertainty quantification methods, thereby forming a valuable assessment benchmark.

A major challenge in structured prediction is to represent the interdependencies within output structures. When outputs are structured as sequences, linear-chain conditional random fields (CRFs) are a widely used model class which can learn local dependencies in the output. However, the CRF's Markov assumption makes it impossible for CRFs to represent distributions with nonlocal dependencies, and standard CRFs are unable to respect nonlocal constraints of the data (such as global arity constraints on output labels). We present a generalization of CRFs that can enforce a broad class of constraints, including nonlocal ones, by specifying the space of possible output structures as a regular language L. The resulting regular-constrained CRF (RegCCRF) has the same formal properties as a standard CRF, but assigns zero probability to all label sequences not in L. Notably, RegCCRFs can incorporate their constraints during training, while related models only enforce constraints during decoding. We prove that constrained training is never worse than constrained decoding, and show empirically that it can be substantially better in practice. Additionally, we demonstrate a practical benefit on downstream tasks by incorporating a RegCCRF into a deep neural model for semantic role labeling, exceeding state-of-the-art results on a standard dataset.

Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

Albeit worryingly underrated in the recent literature on machine learning in general (and, on deep learning in particular), multivariate density estimation is a fundamental task in many applications, at least implicitly, and still an open issue. With a few exceptions, deep neural networks (DNNs) have seldom been applied to density estimation, mostly due to the unsupervised nature of the estimation task, and (especially) due to the need for constrained training algorithms that ended up realizing proper probabilistic models that satisfy Kolmogorov's axioms. Moreover, in spite of the well-known improvement in terms of modeling capabilities yielded by mixture models over plain single-density statistical estimators, no proper mixtures of multivariate DNN-based component densities have been investigated so far. The paper fills this gap by extending our previous work on Neural Mixture Densities (NMMs) to multivariate DNN mixtures. A maximum-likelihood (ML) algorithm for estimating Deep NMMs (DNMMs) is handed out, which satisfies numerically a combination of hard and soft constraints aimed at ensuring satisfaction of Kolmogorov's axioms. The class of probability density functions that can be modeled to any degree of precision via DNMMs is formally defined. A procedure for the automatic selection of the DNMM architecture, as well as of the hyperparameters for its ML training algorithm, is presented (exploiting the probabilistic nature of the DNMM). Experimental results on univariate and multivariate data are reported on, corroborating the effectiveness of the approach and its superiority to the most popular statistical estimation techniques.

Graph neural networks have recently achieved great successes in predicting quantum mechanical properties of molecules. These models represent a molecule as a graph using only the distance between atoms (nodes). They do not, however, consider the spatial direction from one atom to another, despite directional information playing a central role in empirical potentials for molecules, e.g. in angular potentials. To alleviate this limitation we propose directional message passing, in which we embed the messages passed between atoms instead of the atoms themselves. Each message is associated with a direction in coordinate space. These directional message embeddings are rotationally equivariant since the associated directions rotate with the molecule. We propose a message passing scheme analogous to belief propagation, which uses the directional information by transforming messages based on the angle between them. Additionally, we use spherical Bessel functions and spherical harmonics to construct theoretically well-founded, orthogonal representations that achieve better performance than the currently prevalent Gaussian radial basis representations while using fewer than 1/4 of the parameters. We leverage these innovations to construct the directional message passing neural network (DimeNet). DimeNet outperforms previous GNNs on average by 76% on MD17 and by 31% on QM9. Our implementation is available online.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we propose a general method to solve the localization problem by learning a regularization matrix from the localized eigenvectors. Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors representing the global structure. We show applications of our method in several inference problems: community detection in networks, clustering from pairwise similarities, rank estimation and matrix completion problems. Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. This is in contrast with existing spectral algorithms based on data matrix, non-backtracking matrix, Laplacians and those with rank-one regularizations, which perform poorly in the sparse case with noise.

Denoising diffusion probabilistic models (DDPMs) (Ho et al. 2020) have shown impressive results on image and waveform generation in continuous state spaces. Here, we introduce Discrete Denoising Diffusion Probabilistic Models (D3PMs), diffusion-like generative models for discrete data that generalize the multinomial diffusion model of Hoogeboom et al. 2021, by going beyond corruption processes with uniform transition probabilities. This includes corruption with transition matrices that mimic Gaussian kernels in continuous space, matrices based on nearest neighbors in embedding space, and matrices that introduce absorbing states. The third allows us to draw a connection between diffusion models and autoregressive and mask-based generative models. We show that the choice of transition matrix is an important design decision that leads to improved results in image and text domains. We also introduce a new loss function that combines the variational lower bound with an auxiliary cross entropy loss. For text, this model class achieves strong results on character-level text generation while scaling to large vocabularies on LM1B. On the image dataset CIFAR-10, our models approach the sample quality and exceed the log-likelihood of the continuous-space DDPM model.

Currently, high-dimensional data is ubiquitous in data science, which necessitates the development of techniques to decompose and interpret such multidimensional (aka tensor) datasets. Finding a low dimensional representation of the data, that is, its inherent structure, is one of the approaches that can serve to understand the dynamics of low dimensional latent features hidden in the data. Nonnegative RESCAL is one such technique, particularly well suited to analyze self-relational data, such as dynamic networks found in international trade flows. Nonnegative RESCAL computes a low dimensional tensor representation by finding the latent space containing multiple modalities. Estimating the dimensionality of this latent space is crucial for extracting meaningful latent features. Here, to determine the dimensionality of the latent space with nonnegative RESCAL, we propose a latent dimension determination method which is based on clustering of the solutions of multiple realizations of nonnegative RESCAL decompositions. We demonstrate the performance of our model selection method on synthetic data and then we apply our method to decompose a network of international trade flows data from International Monetary Fund and validate the resulting features against empirical facts from economic literature.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ell^2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Non-local (NL) block is a popular module that demonstrates the capability to model global contexts. However, NL block generally has heavy computation and memory costs, so it is impractical to apply the block to high-resolution feature maps. In this paper, to investigate the efficacy of NL block, we empirically analyze if the magnitude and direction of input feature vectors properly affect the attention between vectors. The results show the inefficacy of softmax operation which is generally used to normalize the attention map of the NL block. Attention maps normalized with softmax operation highly rely upon magnitude of key vectors, and performance is degenerated if the magnitude information is removed. By replacing softmax operation with the scaling factor, we demonstrate improved performance on CIFAR-10, CIFAR-100, and Tiny-ImageNet. In Addition, our method shows robustness to embedding channel reduction and embedding weight initialization. Notably, our method makes multi-head attention employable without additional computational cost.

Graph Neural Networks (GNNs), especially message-passing neural networks (MPNNs), have emerged as powerful architectures for learning on graphs in diverse applications. However, MPNNs face challenges when modeling non-local interactions in graphs such as large conjugated molecules, and social networks due to oversmoothing and oversquashing. Although Spectral GNNs and traditional neural networks such as recurrent neural networks and transformers mitigate these challenges, they often lack generalizability, or fail to capture detailed structural relationships or symmetries in the data. To address these concerns, we introduce Matrix Function Neural Networks (MFNs), a novel architecture that parameterizes non-local interactions through analytic matrix equivariant functions. Employing resolvent expansions offers a straightforward implementation and the potential for linear scaling with system size. The MFN architecture achieves stateof-the-art performance in standard graph benchmarks, such as the ZINC and TU datasets, and is able to capture intricate non-local interactions in quantum systems, paving the way to new state-of-the-art force fields.

In recent years, machine learning models like DALL-E, Craiyon, and Stable Diffusion have gained significant attention for their ability to generate high-resolution images from concise descriptions. Concurrently, quantum computing is showing promising advances, especially with quantum machine learning which capitalizes on quantum mechanics to meet the increasing computational requirements of traditional machine learning algorithms. This paper explores the integration of quantum machine learning and variational quantum circuits to augment the efficacy of diffusion-based image generation models. Specifically, we address two challenges of classical diffusion models: their low sampling speed and the extensive parameter requirements. We introduce two quantum diffusion models and benchmark their capabilities against their classical counterparts using MNIST digits, Fashion MNIST, and CIFAR-10. Our models surpass the classical models with similar parameter counts in terms of performance metrics FID, SSIM, and PSNR. Moreover, we introduce a consistency model unitary single sampling architecture that combines the diffusion procedure into a single step, enabling a fast one-step image generation.

The dynamic imbalance of the fore-background is a major challenge in video object counting, which is usually caused by the sparsity of target objects. This remains understudied in existing works and often leads to severe under-/over-prediction errors. To tackle this issue in video object counting, we propose a density-embedded Efficient Masked Autoencoder Counting (E-MAC) framework in this paper. To empower the model's representation ability on density regression, we develop a new Density-Embedded Masked mOdeling (DEMO) method, which first takes the density map as an auxiliary modality to perform multimodal self-representation learning for image and density map. Although DEMO contributes to effective cross-modal regression guidance, it also brings in redundant background information, making it difficult to focus on the foreground regions. To handle this dilemma, we propose an efficient spatial adaptive masking derived from density maps to boost efficiency. Meanwhile, we employ an optical flow-based temporal collaborative fusion strategy to effectively capture the dynamic variations across frames, aligning features to derive multi-frame density residuals. The counting accuracy of the current frame is boosted by harnessing the information from adjacent frames. In addition, considering that most existing datasets are limited to human-centric scenarios, we first propose a large video bird counting dataset, DroneBird, in natural scenarios for migratory bird protection. Extensive experiments on three crowd datasets and our DroneBird validate our superiority against the counterparts. The code and dataset are available.

Diffusion models (DMs) have achieved state-of-the-art results for image synthesis tasks as well as density estimation. Applied in the latent space of a powerful pretrained autoencoder (LDM), their immense computational requirements can be significantly reduced without sacrificing sampling quality. However, DMs and LDMs lack a semantically meaningful representation space as the diffusion process gradually destroys information in the latent variables. We introduce a framework for learning such representations with diffusion models (LRDM). To that end, a LDM is conditioned on the representation extracted from the clean image by a separate encoder. In particular, the DM and the representation encoder are trained jointly in order to learn rich representations specific to the generative denoising process. By introducing a tractable representation prior, we can efficiently sample from the representation distribution for unconditional image synthesis without training of any additional model. We demonstrate that i) competitive image generation results can be achieved with image-parameterized LDMs, ii) LRDMs are capable of learning semantically meaningful representations, allowing for faithful image reconstructions and semantic interpolations. Our implementation is available at https://github.com/jeremiastraub/diffusion.

Estimating the symbolic or analytical form of probability density functions (PDFs) from observed samples is a fundamental challenge in statistical and computational modelling. This process is critical for deriving interpretable and generalizable relationships characterizing the underlying phenomenon. Traditionally, this estimation depends strongly on domain expertise and prior field-specific knowledge, with experts selecting appropriate functional forms or parametric families based on empirical evidence and theoretical understanding. The coefficients of these forms are then typically determined through parameter estimation. In this paper, we develop a framework for estimating symbolic expressions of unnormalized distributions from observed samples using domain-specific prior knowledge, such as the range of interactions and a predefined set of primitive functions. We integrate deep generative models with symbolic regression (SR), incorporating inductive biases, such as factorizing large distributions, to keep the problem tractable. The deep generative models we examine include likelihood-based models, viz., flow models, and score-based models. Experiments show the effectiveness of the proposed framework for estimating density functions for multivariate toy distributions as well as lattices from computational physics, namely, XY model and ϕ^4 theory. When applied to the renormalization problem in ϕ^4 theory, the proposed framework estimates compact symbolic approximations of the hamiltonian function at different scales directly from samples, yielding expressions that may be challenging to derive using traditional perturbative or analytic approaches in nonperturbative settings.

We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time. In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional "flow function". Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals. Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.

Dense retrieval models use bi-encoder network architectures for learning query and document representations. These representations are often in the form of a vector representation and their similarities are often computed using the dot product function. In this paper, we propose a new representation learning framework for dense retrieval. Instead of learning a vector for each query and document, our framework learns a multivariate distribution and uses negative multivariate KL divergence to compute the similarity between distributions. For simplicity and efficiency reasons, we assume that the distributions are multivariate normals and then train large language models to produce mean and variance vectors for these distributions. We provide a theoretical foundation for the proposed framework and show that it can be seamlessly integrated into the existing approximate nearest neighbor algorithms to perform retrieval efficiently. We conduct an extensive suite of experiments on a wide range of datasets, and demonstrate significant improvements compared to competitive dense retrieval models.

Recently, deep clustering methods have gained momentum because of the high representational power of deep neural networks (DNNs) such as autoencoder. The key idea is that representation learning and clustering can reinforce each other: Good representations lead to good clustering while good clustering provides good supervisory signals to representation learning. Critical questions include: 1) How to optimize representation learning and clustering? 2) Should the reconstruction loss of autoencoder be considered always? In this paper, we propose DEKM (for Deep Embedded K-Means) to answer these two questions. Since the embedding space generated by autoencoder may have no obvious cluster structures, we propose to further transform the embedding space to a new space that reveals the cluster-structure information. This is achieved by an orthonormal transformation matrix, which contains the eigenvectors of the within-class scatter matrix of K-means. The eigenvalues indicate the importance of the eigenvectors' contributions to the cluster-structure information in the new space. Our goal is to increase the cluster-structure information. To this end, we discard the decoder and propose a greedy method to optimize the representation. Representation learning and clustering are alternately optimized by DEKM. Experimental results on the real-world datasets demonstrate that DEKM achieves state-of-the-art performance.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Modeling the time-dependent evolution of electron density is essential for understanding quantum mechanical behaviors of condensed matter and enabling predictive simulations in spectroscopy, photochemistry, and ultrafast science. Yet, while machine learning methods have advanced static density prediction, modeling its spatiotemporal dynamics remains largely unexplored. In this work, we introduce a generative framework that combines a 3D convolutional autoencoder with a latent diffusion model (LDM) to learn electron density trajectories from ab-initio molecular dynamics (AIMD) simulations. Our method encodes electron densities into a compact latent space and predicts their future states by sampling from the learned conditional distribution, enabling stable long-horizon rollouts without drift or collapse. To preserve statistical fidelity, we incorporate a scaled Jensen-Shannon divergence regularization that aligns generated and reference density distributions. On AIMD trajectories of liquid lithium at 800 K, our model accurately captures both the spatial correlations and the log-normal-like statistical structure of the density. The proposed framework has the potential to accelerate the simulation of quantum dynamics and overcome key challenges faced by current spatiotemporal machine learning methods as surrogates of quantum mechanical simulators.

Doubly-stochastic matrices (DSM) are increasingly utilized in structure-preserving deep architectures -- such as Optimal Transport layers and Sinkhorn-based attention -- to enforce numerical stability and probabilistic interpretability. In this work, we identify a critical spectral degradation phenomenon inherent to these constraints, termed the Homogeneity Trap. We demonstrate that the maximum-entropy bias, typical of Sinkhorn-based projections, drives the mixing operator towards the uniform barycenter, thereby suppressing the subdominant singular value σ_2 and filtering out high-frequency feature components. We derive a spectral bound linking σ_2 to the network's effective depth, showing that high-entropy constraints restrict feature transformation to a shallow effective receptive field. Furthermore, we formally demonstrate that Layer Normalization fails to mitigate this collapse in noise-dominated regimes; specifically, when spectral filtering degrades the Signal-to-Noise Ratio (SNR) below a critical threshold, geometric structure is irreversibly lost to noise-induced orthogonal collapse. Our findings highlight a fundamental trade-off between entropic stability and spectral expressivity in DSM-constrained networks.

We introduce Latent Space Distribution Matching (LSDM), a novel framework for semi-supervised generative modeling of conditional distributions. LSDM operates in two stages: (i) learning a low-dimensional latent space from both paired and unpaired data, and (ii) performing joint distribution matching in this space via the 1-Wasserstein distance, using only paired data. This two-step approach minimizes an upper bound on the 1-Wasserstein distance between joint distributions, reducing reliance on scarce paired samples while enabling fast one-step generation. Theoretically, we establish non-asymptotic error bounds and demonstrate a key benefit of unpaired data: enhanced geometric fidelity in generated outputs. Furthermore, by extending the scope of its two core steps, LSDM provides a coherent statistical perspective that connects to a broad class of latent-space approaches. Notably, Latent Diffusion Models (LDMs) can be viewed as a variant of LSDM, in which joint distribution matching is achieved indirectly via score matching. Consequently, our results also provide theoretical insights into the consistency of LDMs. Empirical evaluations on real-world image tasks, including class-conditional generation and image super-resolution, demonstrate the effectiveness of LSDM in leveraging unpaired data to enhance generation quality.

In disentangled representation learning, a model is asked to tease apart a dataset's underlying sources of variation and represent them independently of one another. Since the model is provided with no ground truth information about these sources, inductive biases take a paramount role in enabling disentanglement. In this work, we construct an inductive bias towards encoding to and decoding from an organized latent space. Concretely, we do this by (i) quantizing the latent space into discrete code vectors with a separate learnable scalar codebook per dimension and (ii) applying strong model regularization via an unusually high weight decay. Intuitively, the latent space design forces the encoder to combinatorially construct codes from a small number of distinct scalar values, which in turn enables the decoder to assign a consistent meaning to each value. Regularization then serves to drive the model towards this parsimonious strategy. We demonstrate the broad applicability of this approach by adding it to both basic data-reconstructing (vanilla autoencoder) and latent-reconstructing (InfoGAN) generative models. For reliable evaluation, we also propose InfoMEC, a new set of metrics for disentanglement that is cohesively grounded in information theory and fixes well-established shortcomings in previous metrics. Together with regularization, latent quantization dramatically improves the modularity and explicitness of learned representations on a representative suite of benchmark datasets. In particular, our quantized-latent autoencoder (QLAE) consistently outperforms strong methods from prior work in these key disentanglement properties without compromising data reconstruction.

Deep representation learning has become one of the most widely adopted approaches for visual search, recommendation, and identification. Retrieval of such representations from a large database is however computationally challenging. Approximate methods based on learning compact representations, have been widely explored for this problem, such as locality sensitive hashing, product quantization, and PCA. In this work, in contrast to learning compact representations, we propose to learn high dimensional and sparse representations that have similar representational capacity as dense embeddings while being more efficient due to sparse matrix multiplication operations which can be much faster than dense multiplication. Following the key insight that the number of operations decreases quadratically with the sparsity of embeddings provided the non-zero entries are distributed uniformly across dimensions, we propose a novel approach to learn such distributed sparse embeddings via the use of a carefully constructed regularization function that directly minimizes a continuous relaxation of the number of floating-point operations (FLOPs) incurred during retrieval. Our experiments show that our approach is competitive to the other baselines and yields a similar or better speed-vs-accuracy tradeoff on practical datasets.

Since the introduction of deep learning, a wide scope of representation properties, such as decorrelation, whitening, disentanglement, rank, isotropy, and mutual information, have been studied to improve the quality of representation. However, manipulating such properties can be challenging in terms of implementational effectiveness and general applicability. To address these limitations, we propose to regularize von Neumann entropy~(VNE) of representation. First, we demonstrate that the mathematical formulation of VNE is superior in effectively manipulating the eigenvalues of the representation autocorrelation matrix. Then, we demonstrate that it is widely applicable in improving state-of-the-art algorithms or popular benchmark algorithms by investigating domain-generalization, meta-learning, self-supervised learning, and generative models. In addition, we formally establish theoretical connections with rank, disentanglement, and isotropy of representation. Finally, we provide discussions on the dimension control of VNE and the relationship with Shannon entropy. Code is available at: https://github.com/jaeill/CVPR23-VNE.

Despite their many desirable properties, Gaussian processes (GPs) are often compared unfavorably to deep neural networks (NNs) for lacking the ability to learn representations. Recent efforts to bridge the gap between GPs and deep NNs have yielded a new class of inter-domain variational GPs in which the inducing variables correspond to hidden units of a feedforward NN. In this work, we examine some practical issues associated with this approach and propose an extension that leverages the orthogonal decomposition of GPs to mitigate these limitations. In particular, we introduce spherical inter-domain features to construct more flexible data-dependent basis functions for both the principal and orthogonal components of the GP approximation and show that incorporating NN activation features under this framework not only alleviates these shortcomings but is more scalable than alternative strategies. Experiments on multiple benchmark datasets demonstrate the effectiveness of our approach.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

We present Density-Sampled Gaussians (DeG), a novel 3D representation designed to bridge the gap between adaptive rendering primitives and scalable generative modeling. Unlike existing approaches that constrain 3D Gaussians to fixed voxel grids or arrays, DeG models Gaussian centers as samples from a learnable probability density function defined over an octree. This formulation provides a rigorous mathematical framework for adaptive density control: by jointly optimizing the spatial density and Gaussian attributes under rendering supervision, our model naturally concentrates primitives in regions of high geometric complexity. We achieve this via a new render loss contribution gradient that serves as a fully differentiable analogue to the discrete densification and pruning heuristics used in standard Gaussian Splatting. The resulting representation is highly flexible, supporting variable-resolution decoding from a single latent code by simply adjusting the sampling budget. To enable generative synthesis, we train a latent diffusion model on DeG. We identify a critical challenge in applying diffusion to unordered set-structured latents, which can significantly slow convergence, and propose VecSeq, a canonical re-indexing mechanism that anchors latent tokens to a deterministic 3D Sobol sequence. This transforms the ambiguous set-generation problem into a robust sequence modeling task. Extensive experiments demonstrate that our pipeline achieves state-of-the-art quality in single-image-to-3D generation, combining the structural adaptivity of unstructured primitives with the training stability of grid-based methods.

Quantum kernels are among the most promising methods in quantum machine learning, providing a natural way to embed data into high-dimensional Hilbert spaces via quantum circuits and capturing non-linear relationships in data. Their learning behaviour is governed by the function class they represent, which is formally characterized by the structure of the associated Reproducing Kernel Hilbert Space (RKHS). However, a theoretical characterization of this function class, and of how it depends on the choice of feature map and its hyperparameters, is still lacking. In this work, we focus on the Instantaneous Quantum Polynomial (IQP) feature map and derive a Fourier representation of the associated RKHS generators as finite sums of trigonometric functions evaluated on polynomially transformed input variables, with a finite set of accessible frequencies determined by the circuit construction and scaled by the bandwidth hyperparameter. This representation identifies three bandwidth regimes with qualitatively distinct expressive properties from localised, through multimodal, to periodic. Numerical experiments on synthetic datasets with controlled periodic and multimodal structure, across data dimensions and feature-map configurations, reproduce the three regimes in both supervised classification and unsupervised anomaly detection, benchmarked against a classical Gaussian kernel. These results make the relationship between expressivity and data structure explicit for IQP kernels: expressive power translates into a practical benefit only when the accessible frequencies match with the structure of the data, which provides a criterion for selecting the bandwidth when the periodic or multimodal structure of data is known.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

Despite increasing progress in development of methods for generating visual counterfactual explanations, especially with the recent rise of Denoising Diffusion Probabilistic Models, previous works consider them as an entirely local technique. In this work, we take the first step at globalizing them. Specifically, we discover that the latent space of Diffusion Autoencoders encodes the inference process of a given classifier in the form of global directions. We propose a novel proxy-based approach that discovers two types of these directions with the use of only single image in an entirely black-box manner. Precisely, g-directions allow for flipping the decision of a given classifier on an entire dataset of images, while h-directions further increase the diversity of explanations. We refer to them in general as Global Counterfactual Directions (GCDs). Moreover, we show that GCDs can be naturally combined with Latent Integrated Gradients resulting in a new black-box attribution method, while simultaneously enhancing the understanding of counterfactual explanations. We validate our approach on existing benchmarks and show that it generalizes to real-world use-cases.

Machine learning interatomic potentials trained on first-principles reference data are becoming valuable tools for computational physics, biology, and chemistry. Equivariant message-passing neural networks, including transformers, achieve state-of-the-art accuracy but rely on cutoff-based graphs, limiting their ability to capture long-range effects such as electrostatics or dispersion, as well as electron delocalization. While long-range correction schemes based on inverse power laws of interatomic distances have been proposed, they are unable to communicate higher-order geometric information and are thus limited in applicability. To address this shortcoming, we propose the use of equivariant, rather than scalar, charges for long-range interactions, and design a graph neural network architecture, LOREM, around this long-range message passing mechanism. We consider several datasets specifically designed to highlight non-local physical effects, and compare short-range message passing with different receptive fields to invariant and equivariant long-range message passing. Even though most approaches work for careful dataset-specific choices of their model hyperparameters, LOREM works consistently without such changes, with excellent benchmark performance.

We introduce a novel generative model, the Discrete Distribution Networks (DDN), that approximates data distribution using hierarchical discrete distributions. We posit that since the features within a network inherently capture distributional information, enabling the network to generate multiple samples simultaneously, rather than a single output, may offer an effective way to represent distributions. Therefore, DDN fits the target distribution, including continuous ones, by generating multiple discrete sample points. To capture finer details of the target data, DDN selects the output that is closest to the Ground Truth (GT) from the coarse results generated in the first layer. This selected output is then fed back into the network as a condition for the second layer, thereby generating new outputs more similar to the GT. As the number of DDN layers increases, the representational space of the outputs expands exponentially, and the generated samples become increasingly similar to the GT. This hierarchical output pattern of discrete distributions endows DDN with unique properties: more general zero-shot conditional generation and 1D latent representation. We demonstrate the efficacy of DDN and its intriguing properties through experiments on CIFAR-10 and FFHQ. The code is available at https://discrete-distribution-networks.github.io/

To achieve scalable and accurate inference for latent Gaussian processes, we propose a variational approximation based on a family of Gaussian distributions whose covariance matrices have sparse inverse Cholesky (SIC) factors. We combine this variational approximation of the posterior with a similar and efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior. We then focus on a particular SIC ordering and nearest-neighbor-based sparsity pattern resulting in highly accurate prior and posterior approximations. For this setting, our variational approximation can be computed via stochastic gradient descent in polylogarithmic time per iteration. We provide numerical comparisons showing that the proposed double-Kullback-Leibler-optimal Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate for stationary kernels than alternative approaches such as inducing-point and mean-field approximations at similar computational complexity.

Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce ``Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

Deep subspace clustering (DSC) networks based on self-expressive model learn representation matrix, often implemented in terms of fully connected network, in the embedded space. After the learning is finished, representation matrix is used by spectral clustering module to assign labels to clusters. However, such approach ignores complementary information that exist in other layers of the encoder (including the input data themselves). Herein, we apply selected linear subspace clustering algorithm to learn representation matrices from representations learned by all layers of encoder network including the input data. Afterward, we learn a multilayer graph that in a multi-view like manner integrates information from graph Laplacians of all used layers. That improves further performance of selected DSC network. Furthermore, we also provide formulation of our approach to cluster out-of-sample/test data points. We validate proposed approach on four well-known datasets with two DSC networks as baseline models. In almost all the cases, proposed approach achieved statistically significant improvement in three performance metrics. MATLAB code of proposed algorithm is posted on https://github.com/lovro-sinda/MLG-DSC.

Diffusion models trained on different, non-overlapping subsets of a dataset often produce strikingly similar outputs when given the same noise seed. We trace this consistency to a simple linear effect: the shared Gaussian statistics across splits already predict much of the generated images. To formalize this, we develop a random matrix theory (RMT) framework that quantifies how finite datasets shape the expectation and variance of the learned denoiser and sampling map in the linear setting. For expectations, sampling variability acts as a renormalization of the noise level through a self-consistent relation σ^2 mapsto κ(σ^2), explaining why limited data overshrink low-variance directions and pull samples toward the dataset mean. For fluctuations, our variance formulas reveal three key factors behind cross-split disagreement: anisotropy across eigenmodes, inhomogeneity across inputs, and overall scaling with dataset size. Extending deterministic-equivalence tools to fractional matrix powers further allows us to analyze entire sampling trajectories. The theory sharply predicts the behavior of linear diffusion models, and we validate its predictions on UNet and DiT architectures in their non-memorization regime, identifying where and how samples deviates across training data split. This provides a principled baseline for reproducibility in diffusion training, linking spectral properties of data to the stability of generative outputs.

We explore a generative machine learning-based approach for estimating multi-dimensional probability density functions (PDFs) in a target sample using a statistically independent but related control sample - a common challenge in particle physics data analysis. The generative model must accurately reproduce individual observable distributions while preserving the correlations between them, based on the input multidimensional distribution from the control sample. Here we present a conditional normalizing flow model (CNF) based on a chain of bijectors which learns to transform unpaired simulation events to data events. We assess the performance of the CNF model in the context of LHC Higgs to diphoton analysis, where we use the CNF model to convert a Monte Carlo diphoton sample to one that models data. We show that the CNF model can accurately model complex data distributions and correlations. We also leverage the recently popularized Modified Differential Multiplier Method (MDMM) to improve the convergence of our model and assign physical meaning to usually arbitrary loss-function parameters.

Quantum Implicit Neural Representations (QINRs) include components for learning and execution on gate-based quantum computers. While QINRs recently emerged as a promising new paradigm, many challenges concerning their architecture and ansatz design, the utility of quantum-mechanical properties, training efficiency and the interplay with classical modules remain. This paper advances the field by introducing a new type of QINR for 2D image and 3D geometric field learning, which we collectively refer to as Quantum Visual Field (QVF). QVF encodes classical data into quantum statevectors using neural amplitude encoding grounded in a learnable energy manifold, ensuring meaningful Hilbert space embeddings. Our ansatz follows a fully entangled design of learnable parametrised quantum circuits, with quantum (unitary) operations performed in the real Hilbert space, resulting in numerically stable training with fast convergence. QVF does not rely on classical post-processing -- in contrast to the previous QINR learning approach -- and directly employs projective measurement to extract learned signals encoded in the ansatz. Experiments on a quantum hardware simulator demonstrate that QVF outperforms the existing quantum approach and widely used classical foundational baselines in terms of visual representation accuracy across various metrics and model characteristics, such as learning of high-frequency details. We also show applications of QVF in 2D and 3D field completion and 3D shape interpolation, highlighting its practical potential.

We study a setting of collecting and learning from private data distributed across end users. In the shuffled model of differential privacy, the end users partially protect their data locally before sharing it, and their data is also anonymized during its collection to enhance privacy. This model has recently become a prominent alternative to central DP, which requires full trust in a central data curator, and local DP, where fully local data protection takes a steep toll on downstream accuracy. Our main technical result is a shuffled DP protocol for privately estimating the kernel density function of a distributed dataset, with accuracy essentially matching central DP. We use it to privately learn a classifier from the end user data, by learning a private density function per class. Moreover, we show that the density function itself can recover the semantic content of its class, despite having been learned in the absence of any unprotected data. Our experiments show the favorable downstream performance of our approach, and highlight key downstream considerations and trade-offs in a practical ML deployment of shuffled DP.

The use of kernel functions is a common technique to extract important features from data sets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to computational hardness assumptions, cannot be computed classically. It is an important challenge to find quantum kernels that provide an advantage in the classification of real-world data. We introduce a class of quantum kernels that can be used for data with a group structure. The kernel is defined in terms of a unitary representation of the group and a fiducial state that can be optimized using a technique called kernel alignment. We apply this method to a learning problem on a coset-space that embodies the structure of many essential learning problems on groups. We implement the learning algorithm with 27 qubits on a superconducting processor.

Denoising Diffusion Probabilistic Models (DDPMs) achieve state-of-the-art performance in synthesizing new images from random noise, but they lack meaningful latent space that encodes data into features. Recent DDPM-based editing techniques try to mitigate this issue by inverting images back to their approximated staring noise. In this work, we study the relation between the initial Gaussian noise, the samples generated from it, and their corresponding latent encodings obtained through the inversion procedure. First, we interpret their spatial distance relations to show the inaccuracy of the DDIM inversion technique by localizing latent representations manifold between the initial noise and generated samples. Then, we demonstrate the peculiar relation between initial Gaussian noise and its corresponding generations during diffusion training, showing that the high-level features of generated images stabilize rapidly, keeping the spatial distance relationship between noises and generations consistent throughout the training.

Crowd counting is a fundamental problem in crowd analysis which is typically accomplished by estimating a crowd density map and summing over the density values. However, this approach suffers from background noise accumulation and loss of density due to the use of broad Gaussian kernels to create the ground truth density maps. This issue can be overcome by narrowing the Gaussian kernel. However, existing approaches perform poorly when trained with ground truth density maps with broad kernels. To deal with this limitation, we propose using conditional diffusion models to predict density maps, as diffusion models show high fidelity to training data during generation. With that, we present CrowdDiff that generates the crowd density map as a reverse diffusion process. Furthermore, as the intermediate time steps of the diffusion process are noisy, we incorporate a regression branch for direct crowd estimation only during training to improve the feature learning. In addition, owing to the stochastic nature of the diffusion model, we introduce producing multiple density maps to improve the counting performance contrary to the existing crowd counting pipelines. We conduct extensive experiments on publicly available datasets to validate the effectiveness of our method. CrowdDiff outperforms existing state-of-the-art crowd counting methods on several public crowd analysis benchmarks with significant improvements.

It is well noted that coordinate based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been solely studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. Codes available at https://github.com/osiriszjq/Rethinking-positional-encoding.

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.

Despite the non-autoregressive potential of diffusion language models (dLLMs), existing decoding strategies demonstrate positional bias, failing to fully unlock the potential of arbitrary generation. In this work, we delve into the inherent spectral characteristics of dLLMs and present the first frequency-domain analysis showing that low-frequency components in hidden states primarily encode global structural information and long-range dependencies, while high-frequency components are responsible for characterizing local details. Based on this observation, we propose FourierSampler, which leverages a frequency-domain sliding window mechanism to dynamically guide the model to achieve a "structure-to-detail" generation. FourierSampler outperforms other inference enhancement strategies on LLADA and SDAR, achieving relative improvements of 20.4% on LLaDA1.5-8B and 16.0% on LLaDA-8B-Instruct. It notably surpasses similarly sized autoregressive models like Llama3.1-8B-Instruct.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

We propose a new representation of visual data that disentangles object position from appearance. Our method, termed Deep Latent Particles (DLP), decomposes the visual input into low-dimensional latent ``particles'', where each particle is described by its spatial location and features of its surrounding region. To drive learning of such representations, we follow a VAE-based approach and introduce a prior for particle positions based on a spatial-softmax architecture, and a modification of the evidence lower bound loss inspired by the Chamfer distance between particles. We demonstrate that our DLP representations are useful for downstream tasks such as unsupervised keypoint (KP) detection, image manipulation, and video prediction for scenes composed of multiple dynamic objects. In addition, we show that our probabilistic interpretation of the problem naturally provides uncertainty estimates for particle locations, which can be used for model selection, among other tasks. Videos and code are available: https://taldatech.github.io/deep-latent-particles-web/

Generative Adversarial Networks (GANs), especially the recent style-based generators (StyleGANs), have versatile semantics in the structured latent space. Latent semantics discovery methods emerge to move around the latent code such that only one factor varies during the traversal. Recently, an unsupervised method proposed a promising direction to directly use the eigenvectors of the projection matrix that maps latent codes to features as the interpretable directions. However, one overlooked fact is that the projection matrix is non-orthogonal and the number of eigenvectors is too large. The non-orthogonality would entangle semantic attributes in the top few eigenvectors, and the large dimensionality might result in meaningless variations among the directions even if the matrix is orthogonal. To avoid these issues, we propose Householder Projector, a flexible and general low-rank orthogonal matrix representation based on Householder transformations, to parameterize the projection matrix. The orthogonality guarantees that the eigenvectors correspond to disentangled interpretable semantics, while the low-rank property encourages that each identified direction has meaningful variations. We integrate our projector into pre-trained StyleGAN2/StyleGAN3 and evaluate the models on several benchmarks. Within only 1% of the original training steps for fine-tuning, our projector helps StyleGANs to discover more disentangled and precise semantic attributes without sacrificing image fidelity.

Training Large Language Models (LLMs) at ultra-low precision is critically impeded by instability rooted in the conflict between discrete quantization constraints and the intrinsic heavy-tailed spectral nature of linguistic data. By formalizing the connection between Zipfian statistics and random matrix theory, we prove that the power-law decay in the singular value spectra of embeddings is a fundamental requisite for semantic encoding. We derive theoretical bounds showing that uniform quantization introduces a noise floor that disproportionately truncates this spectral tail, which induces spectral flattening and a strictly provable increase in the stable rank of representations. Empirical validation across diverse architectures including GPT-2 and TinyLlama corroborates that this geometric degradation precipitates representational collapse. This work not only quantifies the spectral sensitivity of LLMs but also establishes spectral fidelity as a necessary condition for stable low-bit optimization.

In Open-set Supervised Anomaly Detection (OSAD), the existing methods typically generate pseudo anomalies to compensate for the scarcity of observed anomaly samples, while overlooking critical priors of normal samples, leading to less effective discriminative boundaries. To address this issue, we propose a Distribution Prototype Diffusion Learning (DPDL) method aimed at enclosing normal samples within a compact and discriminative distribution space. Specifically, we construct multiple learnable Gaussian prototypes to create a latent representation space for abundant and diverse normal samples and learn a Schrödinger bridge to facilitate a diffusive transition toward these prototypes for normal samples while steering anomaly samples away. Moreover, to enhance inter-sample separation, we design a dispersion feature learning way in hyperspherical space, which benefits the identification of out-of-distribution anomalies. Experimental results demonstrate the effectiveness and superiority of our proposed DPDL, achieving state-of-the-art performance on 9 public datasets.

Generating new molecules is fundamental to advancing critical applications such as drug discovery and material synthesis. Flows can generate molecules effectively by inverting the encoding process, however, existing flow models either require artifactual dequantization or specific node/edge orderings, lack desiderata such as permutation invariance, or induce discrepancy between the encoding and the decoding steps that necessitates post hoc validity correction. We circumvent these issues with novel continuous normalizing E(3)-equivariant flows, based on a system of node ODEs coupled as a graph PDE, that repeatedly reconcile locally toward globally aligned densities. Our models can be cast as message-passing temporal networks, and result in superlative performance on the tasks of density estimation and molecular generation. In particular, our generated samples achieve state-of-the-art on both the standard QM9 and ZINC250K benchmarks.

Discrete diffusion has recently emerged as a promising paradigm in discrete data modeling. However, existing methods typically rely on a fixed rate transition matrix during training, which not only limits the expressiveness of latent representations, a fundamental strength of variational methods, but also constrains the overall design space. To address these limitations, we propose Discrete Markov Bridge, a novel framework specifically designed for discrete representation learning. Our approach is built upon two key components: Matrix Learning and Score Learning. We conduct a rigorous theoretical analysis, establishing formal performance guarantees for Matrix Learning and proving the convergence of the overall framework. Furthermore, we analyze the space complexity of our method, addressing practical constraints identified in prior studies. Extensive empirical evaluations validate the effectiveness of the proposed Discrete Markov Bridge, which achieves an Evidence Lower Bound (ELBO) of 1.38 on the Text8 dataset, outperforming established baselines. Moreover, the proposed model demonstrates competitive performance on the CIFAR-10 dataset, achieving results comparable to those obtained by image-specific generation approaches.

Deep generative models have been demonstrated as state-of-the-art density estimators. Yet, recent work has found that they often assign a higher likelihood to data from outside the training distribution. This seemingly paradoxical behavior has caused concerns over the quality of the attained density estimates. In the context of hierarchical variational autoencoders, we provide evidence to explain this behavior by out-of-distribution data having in-distribution low-level features. We argue that this is both expected and desirable behavior. With this insight in hand, we develop a fast, scalable and fully unsupervised likelihood-ratio score for OOD detection that requires data to be in-distribution across all feature-levels. We benchmark the method on a vast set of data and model combinations and achieve state-of-the-art results on out-of-distribution detection.

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do not adapt to the structure of a given problem or dataset. In this paper, we aim to represent these bases with neural networks and optimize them. Our key idea is that any target infinite-dimensional orthonormal basis can be viewed either as a point on the Lie manifold of the orthogonal group, or equivalently, as the endpoint of a continuous path on that manifold that connects a reference basis, e.g. Fourier, to that target. Paths on the Lie manifold satisfy ordinary differential equations (ODEs) governed by skew-adjoint integral operators. Using neural networks to define finite-rank generators of such ODEs allows us to parameterize and optimize orthonormal bases in function space. While relying on finite-rank generators to model infinite operators might seem restrictive, we prove a universality result: even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology. In other words, for any target orthonormal basis, there exists a path originating from a reference basis and driven by finite-rank generators that gets arbitrarily close to that target basis. We demonstrate the flexibility of our framework by transforming the Fourier basis into the principal components of a functional dataset, eigenfunctions of linear operators, or dynamic modes of energy-preserving physical simulations.

Quantum theory departs from classical physics in its treatment of correlations, most prominently through the phenomena of contextuality and nonlocality. Once regarded primarily as foundational curiosities, these effects are now understood as key operational resources for quantum computation, communication, and simulation. Although traditionally investigated in distinct settings, recent theoretical and experimental advances have revealed deep conceptual, mathematical, and operational connections between them. This review presents a unified perspective on these developments based on sheaf-theoretic and graph-theoretic frameworks, which provide theory-independent characterizations of statistical correlations. These approaches clarify the structural relationship between contextuality and nonlocality, facilitate the formulation of experimentally testable inequalities, and guide implementations in realistic physical platforms, with particular emphasis on photonic systems. By bridging abstract theoretical structures and concrete experimental realizations, this review sheds light on the nonclassical foundations of quantum correlations and their emerging role in quantum technologies.

We provide an efficient method for computing the maximum likelihood mixed quantum state (with density matrix ρ) given a set of measurement outcome in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix μ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst O(d^4) for the basis change plus O(d^3) for finding ρ where d is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only O(d^3) as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Leveraging nearest neighbor retrieval for self-supervised representation learning has proven beneficial with object-centric images. However, this approach faces limitations when applied to scene-centric datasets, where multiple objects within an image are only implicitly captured in the global representation. Such global bootstrapping can lead to undesirable entanglement of object representations. Furthermore, even object-centric datasets stand to benefit from a finer-grained bootstrapping approach. In response to these challenges, we introduce a novel Cross-Image Object-Level Bootstrapping method tailored to enhance dense visual representation learning. By employing object-level nearest neighbor bootstrapping throughout the training, CrIBo emerges as a notably strong and adequate candidate for in-context learning, leveraging nearest neighbor retrieval at test time. CrIBo shows state-of-the-art performance on the latter task while being highly competitive in more standard downstream segmentation tasks. Our code and pretrained models are publicly available at https://github.com/tileb1/CrIBo.

We present a novel approach for generating minority samples that live on low-density regions of a data manifold. Our framework is built upon diffusion models, leveraging the principle of guided sampling that incorporates an arbitrary energy-based guidance during inference time. The key defining feature of our sampler lies in its self-contained nature, \ie, implementable solely with a pretrained model. This distinguishes our sampler from existing techniques that require expensive additional components (like external classifiers) for minority generation. Specifically, we first estimate the likelihood of features within an intermediate latent sample by evaluating a reconstruction loss w.r.t. its posterior mean. The generation then proceeds with the minimization of the estimated likelihood, thereby encouraging the emergence of minority features in the latent samples of subsequent timesteps. To further improve the performance of our sampler, we provide several time-scheduling techniques that properly manage the influence of guidance over inference steps. Experiments on benchmark real datasets demonstrate that our approach can greatly improve the capability of creating realistic low-likelihood minority instances over the existing techniques without the reliance on costly additional elements. Code is available at https://github.com/soobin-um/sg-minority.

MDPs with low-rank transitions -- that is, the transition matrix can be factored into the product of two matrices, left and right -- is a highly representative structure that enables tractable learning. The left matrix enables expressive function approximation for value-based learning and has been studied extensively. In this work, we instead investigate sample-efficient learning with density features, i.e., the right matrix, which induce powerful models for state-occupancy distributions. This setting not only sheds light on leveraging unsupervised learning in RL, but also enables plug-in solutions for convex RL. In the offline setting, we propose an algorithm for off-policy estimation of occupancies that can handle non-exploratory data. Using this as a subroutine, we further devise an online algorithm that constructs exploratory data distributions in a level-by-level manner. As a central technical challenge, the additive error of occupancy estimation is incompatible with the multiplicative definition of data coverage. In the absence of strong assumptions like reachability, this incompatibility easily leads to exponential error blow-up, which we overcome via novel technical tools. Our results also readily extend to the representation learning setting, when the density features are unknown and must be learned from an exponentially large candidate set.

Identifying Out-of-distribution (OOD) data is becoming increasingly critical as the real-world applications of deep learning methods expand. Post-hoc methods modify softmax scores fine-tuned on outlier data or leverage intermediate feature layers to identify distinctive patterns between In-Distribution (ID) and OOD samples. Other methods focus on employing diverse OOD samples to learn discrepancies between ID and OOD. These techniques, however, are typically dependent on the quality of the outlier samples assumed. Density-based methods explicitly model class-conditioned distributions but this requires long training time or retraining the classifier. To tackle these issues, we introduce FlowCon, a new density-based OOD detection technique. Our main innovation lies in efficiently combining the properties of normalizing flow with supervised contrastive learning, ensuring robust representation learning with tractable density estimation. Empirical evaluation shows the enhanced performance of our method across common vision datasets such as CIFAR-10 and CIFAR-100 pretrained on ResNet18 and WideResNet classifiers. We also perform quantitative analysis using likelihood plots and qualitative visualization using UMAP embeddings and demonstrate the robustness of the proposed method under various OOD contexts. Code will be open-sourced post decision.

Modeling long-range dependencies in time series generation poses a fundamental trade-off between representational capacity and computational efficiency. Traditional temporal diffusion models suffer from local entanglement and the O(L^2) cost of attention mechanisms. We address these limitations by introducing PaCoDi (Parallel Complex Diffusion), a spectral-native architecture that decouples generative modeling in the frequency domain. PaCoDi fundamentally alters the problem topology: the Fourier Transform acts as a diagonalizing operator, converting locally coupled temporal signals into globally decorrelated spectral components. Theoretically, we prove the Quadrature Forward Diffusion and Conditional Reverse Factorization theorem, demonstrating that the complex diffusion process can be split into independent real and imaginary branches. We bridge the gap between this decoupled theory and data reality using a Mean Field Theory (MFT) approximation reinforced by an interactive correction mechanism. Furthermore, we generalize this discrete DDPM to continuous-time Frequency SDEs, rigorously deriving the Spectral Wiener Process describe the differential spectral Brownian motion limit. Crucially, PaCoDi exploits the Hermitian Symmetry of real-valued signals to compress the sequence length by half, achieving a 50% reduction in attention FLOPs without information loss. We further derive a rigorous Heteroscedastic Loss to handle the non-isotropic noise distribution on the compressed manifold. Extensive experiments show that PaCoDi outperforms existing baselines in both generation quality and inference speed, offering a theoretically grounded and computationally efficient solution for time series modeling.

Quantum kernels hold great promise for offering computational advantages over classical learners, with the effectiveness of these kernels closely tied to the design of the quantum feature map. However, the challenge of designing effective quantum feature maps for real-world datasets, particularly in the absence of sufficient prior information, remains a significant obstacle. In this study, we present a data-driven approach that automates the design of problem-specific quantum feature maps. Our approach leverages feature-selection techniques to handle high-dimensional data on near-term quantum machines with limited qubits, and incorporates a deep neural predictor to efficiently evaluate the performance of various candidate quantum kernels. Through extensive numerical simulations on different datasets, we demonstrate the superiority of our proposal over prior methods, especially for the capability of eliminating the kernel concentration issue and identifying the feature map with prediction advantages. Our work not only unlocks the potential of quantum kernels for enhancing real-world tasks but also highlights the substantial role of deep learning in advancing quantum machine learning.

Attention operators have been applied on both 1-D data like texts and higher-order data such as images and videos. Use of attention operators on high-order data requires flattening of the spatial or spatial-temporal dimensions into a vector, which is assumed to follow a multivariate normal distribution. This not only incurs excessive requirements on computational resources, but also fails to preserve structures in data. In this work, we propose to avoid flattening by assuming the data follow matrix-variate normal distributions. Based on this new view, we develop Kronecker attention operators (KAOs) that operate on high-order tensor data directly. More importantly, the proposed KAOs lead to dramatic reductions in computational resources. Experimental results show that our methods reduce the amount of required computational resources by a factor of hundreds, with larger factors for higher-dimensional and higher-order data. Results also show that networks with KAOs outperform models without attention, while achieving competitive performance as those with original attention operators.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Contemporary models of high dimensional physical systems are constrained by the curse of dimensionality and a reliance on dense data. We introduce KHRONOS (Kernel Expansion Hierarchy for Reduced Order, Neural Optimized Surrogates), an AI framework for model based, model free and model inversion tasks. KHRONOS constructs continuously differentiable target fields with a hierarchical composition of per-dimension kernel expansions, which are tensorized into modes and then superposed. We evaluate KHRONOS on a canonical 2D, Poisson equation benchmark: across 16 to 512 degrees of freedom (DoFs), it obtained L_2-square errors of 5e-4 down to 6e-11. This represents a greater than 100-fold gain over Kolmogorov Arnold Networks (which itself reports a 100 times improvement on MLPs/PINNs with 100 times fewer parameters) when controlling for the number of parameters. This also represents a 1e6-fold improvement in L_2-square error compared to standard linear FEM at comparable DoFs. Inference complexity is dominated by inner products, yielding sub-millisecond full-field predictions that scale to an arbitrary resolution. For inverse problems, KHRONOS facilitates rapid, iterative level set recovery in only a few forward evaluations, with sub-microsecond per sample latency. KHRONOS's scalability, expressivity, and interpretability open new avenues in constrained edge computing, online control, computer vision, and beyond.

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

Flow matching has emerged as a powerful generative modeling approach with flexible choices of source distribution. While Gaussian distributions are commonly used, the potential for better alternatives in high-dimensional data generation remains largely unexplored. In this paper, we propose a novel 2D simulation that captures high-dimensional geometric properties in an interpretable 2D setting, enabling us to analyze the learning dynamics of flow matching during training. Based on this analysis, we derive several key insights about flow matching behavior: (1) density approximation can paradoxically degrade performance due to mode discrepancy, (2) directional alignment suffers from path entanglement when overly concentrated, (3) Gaussian's omnidirectional coverage ensures robust learning, and (4) norm misalignment incurs substantial learning costs. Building on these insights, we propose a practical framework that combines norm-aligned training with directionally-pruned sampling. This approach maintains the robust omnidirectional supervision essential for stable flow learning, while eliminating initializations in data-sparse regions during inference. Importantly, our pruning strategy can be applied to any flow matching model trained with a Gaussian source, providing immediate performance gains without the need for retraining. Empirical evaluations demonstrate consistent improvements in both generation quality and sampling efficiency. Our findings provide practical insights and guidelines for source distribution design and introduce a readily applicable technique for improving existing flow matching models. Our code is available at https://github.com/kwanseokk/SourceFM.

Clustering aims to group unlabelled samples based on their similarities. It has become a significant tool for the analysis of high-dimensional data. However, most of the clustering methods merely generate pseudo labels and thus are unable to simultaneously present the similarities between different clusters and outliers. This paper proposes a new framework called High-dimensional Clustering onto Hamiltonian Cycle (HCHC) to solve the above problems. First, HCHC combines global structure with local structure in one objective function for deep clustering, improving the labels as relative probabilities, to mine the similarities between different clusters while keeping the local structure in each cluster. Then, the anchors of different clusters are sorted on the optimal Hamiltonian cycle generated by the cluster similarities and mapped on the circumference of a circle. Finally, a sample with a higher probability of a cluster will be mapped closer to the corresponding anchor. In this way, our framework allows us to appreciate three aspects visually and simultaneously - clusters (formed by samples with high probabilities), cluster similarities (represented as circular distances), and outliers (recognized as dots far away from all clusters). The experiments illustrate the superiority of HCHC.

A framework is proposed that addresses both conditional density estimation and latent variable discovery. The objective function maximizes explanation of variability in the data, achieved through the optimal transport barycenter generalized to a collection of conditional distributions indexed by a covariate --either given or latent-- in any suitable space. Theoretical results establish the existence of barycenters, a minimax formulation of optimal transport maps, and a general characterization of variability via the optimal transport cost. This framework leads to a family of non-parametric neural network-based algorithms, the BaryNet, with a supervised version that estimates conditional distributions and an unsupervised version that assigns latent variables. The efficacy of BaryNets is demonstrated by tests on both artificial and real-world data sets. A parallel drawn between autoencoders and the barycenter framework leads to the Barycentric autoencoder algorithm (BAE).

Denoising Diffusion Probabilistic Models (DDPMs) show promising potential in one-step Real-World Image Super-Resolution (Real-ISR). Current one-step Real-ISR methods typically inject the low-quality (LQ) image latent representation at the start or end timestep of the DDPM scheduler. Recent studies have begun to note that the LQ image latent and the pre-trained noisy latent representations are intuitively closer at a mid-timestep. However, a quantitative analysis of these latent representations remains lacking. Considering these latent representations can be decomposed into signal and noise, we propose a method based on the Signal-to-Noise Ratio (SNR) to pre-compute an average optimal mid-timestep for injection. To better approximate the pre-trained noisy latent representation, we further introduce the Latent Representation Refinement (LRR) loss via a LoRA-enhanced VAE encoder. We also fine-tune the backbone of the DDPM-based generative model using LoRA to perform one-step denoising at the average optimal mid-timestep. Based on these components, we present OMGSR, a GAN-based Real-ISR framework that employs a DDPM-based generative model as the generator and a DINOv3-ConvNeXt model with multi-level discriminator heads as the discriminator. We also propose the DINOv3-ConvNeXt DISTS (Dv3CD) loss, which is enhanced for structural perception at varying resolutions. Within the OMGSR framework, we develop OMGSR-S based on SD2.1-base. An ablation study confirms that our pre-computation strategy and LRR loss significantly improve the baseline. Comparative studies demonstrate that OMGSR-S achieves state-of-the-art performance across multiple metrics. Code is available at Github{https://github.com/wuer5/OMGSR}.

Given time series data, how can we answer questions like "what will happen in the future?" and "how did we get here?" These sorts of probabilistic inference questions are challenging when observations are high-dimensional. In this paper, we show how these questions can have compact, closed form solutions in terms of learned representations. The key idea is to apply a variant of contrastive learning to time series data. Prior work already shows that the representations learned by contrastive learning encode a probability ratio. By extending prior work to show that the marginal distribution over representations is Gaussian, we can then prove that joint distribution of representations is also Gaussian. Taken together, these results show that representations learned via temporal contrastive learning follow a Gauss-Markov chain, a graphical model where inference (e.g., prediction, planning) over representations corresponds to inverting a low-dimensional matrix. In one special case, inferring intermediate representations will be equivalent to interpolating between the learned representations. We validate our theory using numerical simulations on tasks up to 46-dimensions.

In this paper, we show that a binary latent space can be explored for compact yet expressive image representations. We model the bi-directional mappings between an image and the corresponding latent binary representation by training an auto-encoder with a Bernoulli encoding distribution. On the one hand, the binary latent space provides a compact discrete image representation of which the distribution can be modeled more efficiently than pixels or continuous latent representations. On the other hand, we now represent each image patch as a binary vector instead of an index of a learned cookbook as in discrete image representations with vector quantization. In this way, we obtain binary latent representations that allow for better image quality and high-resolution image representations without any multi-stage hierarchy in the latent space. In this binary latent space, images can now be generated effectively using a binary latent diffusion model tailored specifically for modeling the prior over the binary image representations. We present both conditional and unconditional image generation experiments with multiple datasets, and show that the proposed method performs comparably to state-of-the-art methods while dramatically improving the sampling efficiency to as few as 16 steps without using any test-time acceleration. The proposed framework can also be seamlessly scaled to 1024 times 1024 high-resolution image generation without resorting to latent hierarchy or multi-stage refinements.

Contrastively trained encoders have recently been proven to invert the data-generating process: they encode each input, e.g., an image, into the true latent vector that generated the image (Zimmermann et al., 2021). However, real-world observations often have inherent ambiguities. For instance, images may be blurred or only show a 2D view of a 3D object, so multiple latents could have generated them. This makes the true posterior for the latent vector probabilistic with heteroscedastic uncertainty. In this setup, we extend the common InfoNCE objective and encoders to predict latent distributions instead of points. We prove that these distributions recover the correct posteriors of the data-generating process, including its level of aleatoric uncertainty, up to a rotation of the latent space. In addition to providing calibrated uncertainty estimates, these posteriors allow the computation of credible intervals in image retrieval. They comprise images with the same latent as a given query, subject to its uncertainty. Code is available at https://github.com/mkirchhof/Probabilistic_Contrastive_Learning

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.

Neural tangent kernels (NTKs) are a powerful tool for analyzing deep, non-linear neural networks. In the infinite-width limit, NTKs can easily be computed for most common architectures, yielding full analytic control over the training dynamics. However, at infinite width, important properties of training such as NTK evolution or feature learning are absent. Nevertheless, finite width effects can be included by computing corrections to the Gaussian statistics at infinite width. We introduce Feynman diagrams for computing finite-width corrections to NTK statistics. These dramatically simplify the necessary algebraic manipulations and enable the computation of layer-wise recursion relations for arbitrary statistics involving preactivations, NTKs and certain higher-derivative tensors (dNTK and ddNTK) required to predict the training dynamics at leading order. We demonstrate the feasibility of our framework by extending stability results for deep networks from preactivations to NTKs and proving the absence of finite-width corrections for scale-invariant nonlinearities such as ReLU on the diagonal of the Gram matrix of the NTK. We numerically implement the complete set of equations necessary to compute the first-order corrections for arbitrary inputs and demonstrate that the results follow the statistics of sampled neural networks for widths ngtrsim 20.

Machine-learning generative methods for material design are constructed by representing a given chemical structure, either a solid or a molecule, over appropriate atomic features, generally called structural descriptors. These must be fully descriptive of the system, must facilitate the training process and must be invertible, so that one can extract the atomic configurations corresponding to the output of the model. In general, this last requirement is not automatically satisfied by the most efficient structural descriptors, namely the representation is not directly invertible. Such drawback severely limits our freedom of choice in selecting the most appropriate descriptors for the problem, and thus our flexibility to construct generative models. In this work, we present a general optimization method capable of inverting any local many-body descriptor of the chemical environment, back to a cartesian representation. The algorithm is then implemented together with the bispectrum representation of the local structure and demonstrated for a number of molecules. The scheme presented here, thus, represents a general approach to the inversion of structural descriptors, enabling the construction of efficient structural generative models.

Standard sequence models encode position by token rank rather than physical coordinate, typically forcing practitioners to destructively resample continuous physical signals such as spectra observed across heterogeneous instruments. We identify this coordinate mismatch as a structural limitation in cross-instrument representation learning, and propose OmniSpectra, a wavelength-native foundation model for spectral representation learning: a continuous-coordinate masked autoencoder that discards the integer pixel indexing of sequence modeling. Mapping physical coordinates to a logarithmic basis turns the multiplicative scaling induced by cosmological redshift into a uniform additive shift. In continuous rotary position embeddings, this shared shift cancels from relative token distances rather than producing grid-dependent positional distortion. We further counter the spectral bias inherent to implicit neural decoders with a physics-grounded, resolution-gated Fourier decoder that provides anti-aliasing. Pretrained on eight surveys and benchmarked against seven baselines, our model reaches state-of-the-art performance with only 12.2M parameters while outperforming substantially larger models, indicating that encoding physical structure can substitute for parameter scale.

We study the emergence of multi-step reasoning in deep Transformer language models through a geometric and statistical-physics lens. Treating the hidden-state trajectory as a flow on an implicit Riemannian manifold, we analyze the layerwise covariance spectrum of activations, where C^{(ell)}=E[h^{(ell)}h^{(ell)top}], and track deviations from a random-matrix bulk. Across model scales (1.5B--30B), we observe a sharp reduction in effective dimensionality consistent with a phase transition: an order parameter based on sparsity/localization, Ω(h)=1-|h|_1/(d|h|_2), exhibits a discontinuity near a critical normalized depth γ_capprox 0.42 in sufficiently large models. We formalize the forward pass as a discrete coarse-graining map and relate the appearance of stable "concept basins" to fixed points of this renormalization-like dynamics. The resulting low-entropy regime is characterized by a spectral tail collapse and by the formation of transient, reusable object-like structures in representation space, which we call Transient Class Objects (TCOs). We provide theoretical conditions connecting logical separability to spectral decay and validate the predicted signatures with layerwise probes on multiple open-weight model families.

Diffusion models have shown remarkable performance on many generative tasks. Despite recent success, most diffusion models are restricted in that they only allow linear transformation of the data distribution. In contrast, broader family of transformations can potentially help train generative distributions more efficiently, simplifying the reverse process and closing the gap between the true negative log-likelihood and the variational approximation. In this paper, we present Neural Diffusion Models (NDMs), a generalization of conventional diffusion models that enables defining and learning time-dependent non-linear transformations of data. We show how to optimise NDMs using a variational bound in a simulation-free setting. Moreover, we derive a time-continuous formulation of NDMs, which allows fast and reliable inference using off-the-shelf numerical ODE and SDE solvers. Finally, we demonstrate the utility of NDMs with learnable transformations through experiments on standard image generation benchmarks, including CIFAR-10, downsampled versions of ImageNet and CelebA-HQ. NDMs outperform conventional diffusion models in terms of likelihood and produce high-quality samples.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

Registration of distant outdoor LiDAR point clouds is crucial to extending the 3D vision of collaborative autonomous vehicles, and yet is challenging due to small overlapping area and a huge disparity between observed point densities. In this paper, we propose Group-wise Contrastive Learning (GCL) scheme to extract density-invariant geometric features to register distant outdoor LiDAR point clouds. We mark through theoretical analysis and experiments that, contrastive positives should be independent and identically distributed (i.i.d.), in order to train densityinvariant feature extractors. We propose upon the conclusion a simple yet effective training scheme to force the feature of multiple point clouds in the same spatial location (referred to as positive groups) to be similar, which naturally avoids the sampling bias introduced by a pair of point clouds to conform with the i.i.d. principle. The resulting fully-convolutional feature extractor is more powerful and density-invariant than state-of-the-art methods, improving the registration recall of distant scenarios on KITTI and nuScenes benchmarks by 40.9% and 26.9%, respectively. Code is available at https://github.com/liuQuan98/GCL.

Dense Associative Memories (DenseAMs) are generalizations of Hopfield networks, which have superior information storage capacity and can store training data points (memories) at local minima of the energy landscape. When the amount of training data exceeds the critical memory storage capacity of these models, new local minima, which are different from the training data, emerge. In Associative Memory these emergent local minima are called spurious; states, which hinder memory retrieval. In this work, we examine diffusion models (DMs) through the DenseAM lens, viewing their generative process as an attempt of a memory retrieval. In the small data regimes, DMs create distinct attractors for each training sample, akin to DenseAMs below the critical memory storage. As the training data size increases, they transition from memorization to generalization. We identify a critical intermediate phase, predicted by DenseAM theory -- the spurious states. In generative modeling, these states are no longer negative artifacts but rather are the first signs of generative capabilities. We characterize the basins of attraction, energy landscape curvature, and computational properties of these previously overlooked states. Their existence is demonstrated across a wide range of architectures and datasets.

Physics is based on probabilities as fundamental entities of a mathematical description. Expectation values of observables are computed according to the classical statistical rule. The overall probability distribution for one world covers all times. The quantum formalism arises once one focuses on the evolution of the time-local probabilistic information. Wave functions or the density matrix allow the formulation of a general linear evolution law for classical statistics. The quantum formalism for classical statistics is a powerful tool which allows us to implement for generalized Ising models the momentum observable with the associated Fourier representation. The association of operators to observables permits the computation of expectation values in terms of the density matrix by the usual quantum rule. We show that probabilistic cellular automata are quantum systems in a formulation with discrete time steps and real wave functions. With a complex structure the evolution operator for automata can be expressed in terms of a Hamiltonian involving fermionic creation and annihilation operators. The time-local probabilistic information amounts to a subsystem of the overall probabilistic system which is correlated with its environment consisting of the past and future. Such subsystems typically involve probabilistic observables for which only a probability distribution for their possible measurement values is available. Incomplete statistics does not permit to compute classical correlation functions for arbitrary subsystem-observables. Bell's inequalities are not generally applicable.

Combinatorial inverse problems in high energy physics span enormous algorithmic challenges. This work presents a new deep learning driven clustering algorithm that utilizes a space-time non-local trainable graph constructor, a graph neural network, and a set transformer. The model is trained with loss functions at the graph node, edge and object level, including contrastive learning and meta-supervision. The algorithm can be applied to problems such as charged particle tracking, calorimetry, pile-up discrimination, jet physics, and beyond. We showcase the effectiveness of this cutting-edge AI approach through particle tracking simulations. The code is available online.

We present a principled approach to uncover the structure of visual data by solving a novel deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Unfortunately, permutation matrices are discrete, thereby posing difficulties for gradient-based methods. To this end, we resort to a continuous approximation of these matrices using doubly-stochastic matrices which we generate from standard CNN predictions using Sinkhorn iterations. Unrolling these iterations in a Sinkhorn network layer, we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on two challenging computer vision problems, namely, (i) relative attributes learning and (ii) self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for (i) and on the classification and segmentation tasks on the PASCAL VOC dataset for (ii).

Classical deep neural networks can learn rich multi-particle correlations in collider data, but their inductive biases are rarely anchored in physics structure. We propose quantum-informed neural networks (QINNs), a general framework that brings quantum information concepts and quantum observables into purely classical models. While the framework is broad, in this paper, we study one concrete realisation that encodes each particle as a qubit and uses the Quantum Fisher Information Matrix (QFIM) as a compact, basis-independent summary of particle correlations. Using jet tagging as a case study, QFIMs act as lightweight embeddings in graph neural networks, increasing model expressivity and plasticity. The QFIM reveals distinct patterns for QCD and hadronic top jets that align with physical expectations. Thus, QINNs offer a practical, interpretable, and scalable route to quantum-informed analyses, that is, tomography, of particle collisions, particularly by enhancing well-established deep learning approaches.

Cascaded models are multi-scale generative models with a marked capacity for producing perceptually impressive samples at high resolutions. In this work, we show that they can also be excellent likelihood models, so long as we overcome a fundamental difficulty with probabilistic multi-scale models: the intractability of the likelihood function. Chiefly, in cascaded models each intermediary scale introduces extraneous variables that cannot be tractably marginalized out for likelihood evaluation. This issue vanishes by modeling the diffusion process on latent spaces induced by a class of transformations we call hierarchical volume-preserving maps, which decompose spatially structured data in a hierarchical fashion without introducing local distortions in the latent space. We demonstrate that two such maps are well-known in the literature for multiscale modeling: Laplacian pyramids and wavelet transforms. Not only do such reparameterizations allow the likelihood function to be directly expressed as a joint likelihood over the scales, we show that the Laplacian pyramid and wavelet transform also produces significant improvements to the state-of-the-art on a selection of benchmarks in likelihood modeling, including density estimation, lossless compression, and out-of-distribution detection. Investigating the theoretical basis of our empirical gains we uncover deep connections to score matching under the Earth Mover's Distance (EMD), which is a well-known surrogate for perceptual similarity. Code can be found at https://github.com/lihenryhfl/pcdm{this https url}.

Out-of-distribution (OOD) detection is a critical requirement for the deployment of deep neural networks. This paper introduces the HEAT model, a new post-hoc OOD detection method estimating the density of in-distribution (ID) samples using hybrid energy-based models (EBM) in the feature space of a pre-trained backbone. HEAT complements prior density estimators of the ID density, e.g. parametric models like the Gaussian Mixture Model (GMM), to provide an accurate yet robust density estimation. A second contribution is to leverage the EBM framework to provide a unified density estimation and to compose several energy terms. Extensive experiments demonstrate the significance of the two contributions. HEAT sets new state-of-the-art OOD detection results on the CIFAR-10 / CIFAR-100 benchmark as well as on the large-scale Imagenet benchmark. The code is available at: https://github.com/MarcLafon/heatood.

Evaluation of spatiotemporal point process (STPP) models relies heavily on opaque real-world datasets, where latent generative structure is unknown and model failures are difficult to attribute. We introduce HawkesNest, a generator-aligned benchmark for controlled spatiotemporal pattern complexity built on a multivariate Hawkes backbone. HawkesNest defines four complexity axes: space--time entanglement, background heterogeneity, cross-type interaction, and domain topology. Each axis is associated with a deterministic index computed from the latent data-generating mechanism. By varying these axes while holding global rate, stability, and simulation budget fixed, HawkesNest enables diagnostic stress tests of STPP models under known structural difficulty. We verify that the indices are monotone and nearly orthogonal under controlled sweeps. We illustrate its use by showing that Hawkes-family baselines degrade under joint heterogeneity--entanglement complexity, even though they are structurally aligned with the Hawkes data-generating backbone. We further show that HawkesNest exposes neural-model sensitivity: AutoSTPP remains vulnerable under isolated increases in space--time entanglement. Code. Available at https://github.com/YahyaAalaila/HawkesNest

Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer vision, and time series problems. Neural operators, which map spaces of functions into spaces of functions, are necessarily both nonlinear and nonlocal if they are universal; it is thus natural to ask whether the attention mechanism can be used in the design of neural operators. Motivated by this, we study transformers in the function space setting. We formulate attention as a map between infinite dimensional function spaces and prove that the attention mechanism as implemented in practice is a Monte Carlo or finite difference approximation of this operator. The function space formulation allows for the design of transformer neural operators, a class of architectures designed to learn mappings between function spaces. In this paper, we state and prove the first universal approximation result for transformer neural operators, using only a slight modification of the architecture implemented in practice. The prohibitive cost of applying the attention operator to functions defined on multi-dimensional domains leads to the need for more efficient attention-based architectures. For this reason we also introduce a function space generalization of the patching strategy from computer vision, and introduce a class of associated neural operators. Numerical results, on an array of operator learning problems, demonstrate the promise of our approaches to function space formulations of attention and their use in neural operators.

We give a classical analogue to Kerenidis and Prakash's quantum recommendation system, previously believed to be one of the strongest candidates for provably exponential speedups in quantum machine learning. Our main result is an algorithm that, given an m times n matrix in a data structure supporting certain ell^2-norm sampling operations, outputs an ell^2-norm sample from a rank-k approximation of that matrix in time O(poly(k)log(mn)), only polynomially slower than the quantum algorithm. As a consequence, Kerenidis and Prakash's algorithm does not in fact give an exponential speedup over classical algorithms. Further, under strong input assumptions, the classical recommendation system resulting from our algorithm produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n. The main insight of this work is the use of simple routines to manipulate ell^2-norm sampling distributions, which play the role of quantum superpositions in the classical setting. This correspondence indicates a potentially fruitful framework for formally comparing quantum machine learning algorithms to classical machine learning algorithms.

An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the multidimensional torus is analyzed. The equations describe the dynamics of population species with repulsive or attractive interactions. The numerical scheme is based on a generalized Scharfetter-Gummel discretization of the nonlocal flux term. For merely integrable kernel functions, the scheme preserves the positivity, total mass, and entropy structure. The existence of a discrete solution and its convergence to a solution to the continuous problem, as the mesh size tends to zero, are shown. A key difficulty is the degeneracy of the generalized Bernoulli function in the Scharfetter-Gummel approximation. This issue is overcome by proving a uniform estimate for the discrete Fisher information, which requires both the Boltzmann and Rao entropy inequalities. Numerical simulations illustrate the features of the scheme in one and two space dimensions.

In this research, we explore the integration of quantum computing with classical machine learning for image classification tasks, specifically focusing on the MNIST dataset. We propose a hybrid quantum-classical approach that leverages the strengths of both paradigms. The process begins with preprocessing the MNIST dataset, normalizing the pixel values, and reshaping the images into vectors. An autoencoder compresses these 784-dimensional vectors into a 64-dimensional latent space, effectively reducing the data's dimensionality while preserving essential features. These compressed features are then processed using a quantum circuit implemented on a 5-qubit system. The quantum circuit applies rotation gates based on the feature values, followed by Hadamard and CNOT gates to entangle the qubits, and measurements are taken to generate quantum outcomes. These outcomes serve as input for a classical neural network designed to classify the MNIST digits. The classical neural network comprises multiple dense layers with batch normalization and dropout to enhance generalization and performance. We evaluate the performance of this hybrid model and compare it with a purely classical approach. The experimental results indicate that while the hybrid model demonstrates the feasibility of integrating quantum computing with classical techniques, the accuracy of the final model, trained on quantum outcomes, is currently lower than the classical model trained on compressed features. This research highlights the potential of quantum computing in machine learning, though further optimization and advanced quantum algorithms are necessary to achieve superior performance.

Implicit Neural Representations (INRs) have emerged as a powerful paradigm for representing signals such as images, audio, and 3D scenes. However, existing INR frameworks -- including MLPs with Fourier features, SIREN, and multiresolution hash grids -- implicitly assume a global and stationary spectral basis. This assumption is fundamentally misaligned with real-world signals whose frequency characteristics vary significantly across space, exhibiting local high-frequency textures, smooth regions, and frequency drift phenomena. We propose Neural Spectral Transport Representation (NSTR), the first INR framework that explicitly models a spatially varying local frequency field. NSTR introduces a learnable frequency transport equation, a PDE that governs how local spectral compositions evolve across space. Given a learnable local spectrum field S(x) and a frequency transport network F_θ enforcing nabla S(x) approx F_θ(x, S(x)), NSTR reconstructs signals by spatially modulating a compact set of global sinusoidal bases. This formulation enables strong local adaptivity and offers a new level of interpretability via visualizing frequency flows. Experiments on 2D image regression, audio reconstruction, and implicit 3D geometry show that NSTR achieves significantly better accuracy-parameter trade-offs than SIREN, Fourier-feature MLPs, and Instant-NGP. NSTR requires fewer global frequencies, converges faster, and naturally explains signal structure through spectral transport fields. We believe NSTR opens a new direction in INR research by introducing explicit modeling of space-varying spectrum.

Causal representation learning has showed a variety of settings in which we can disentangle latent variables with identifiability guarantees (up to some reasonable equivalence class). Common to all of these approaches is the assumption that (1) the latent variables are represented as d-dimensional vectors, and (2) that the observations are the output of some injective generative function of these latent variables. While these assumptions appear benign, we show that when the observations are of multiple objects, the generative function is no longer injective and disentanglement fails in practice. We can address this failure by combining recent developments in object-centric learning and causal representation learning. By modifying the Slot Attention architecture arXiv:2006.15055, we develop an object-centric architecture that leverages weak supervision from sparse perturbations to disentangle each object's properties. This approach is more data-efficient in the sense that it requires significantly fewer perturbations than a comparable approach that encodes to a Euclidean space and we show that this approach successfully disentangles the properties of a set of objects in a series of simple image-based disentanglement experiments.

Hyper-Connections (HC) generalize residual connections into multiple streams, employing residual matrices for cross-stream feature mixing to enrich model expressivity. However, unconstrained mixing disrupts the identity mapping property intrinsic to the residual connection, causing unstable training. To address this, Manifold-Constrained Hyper-Connections (mHC) and its variant restrict these matrices to the Birkhoff polytope (doubly stochastic matrices) via Sinkhorn iterations or permutation-based parameterizations. We reveal three limitations of this polytope constraint: (1) identity degeneration, where learned matrices collapse around the identity and diminish cross-stream interactions, (2) an expressivity bottleneck, as the non-negativity constraint prevents subtractive feature disentanglement, and (3) parameterization inefficiencies, manifesting as unstable Sinkhorn iterations or the factorial-scaling overhead of permutation-based parameterizations. To overcome these flaws, we propose Spectral-Sphere-Constrained Hyper-Connections (sHC). By geometrically shifting the feasible set from a rigid polytope to a spectral norm sphere, sHC allows negative entries, unlocking subtractive interactions for selective feature diversification. This shift eliminates unstable Sinkhorn projections and factorial parameterization, enabling expressive, non-degenerate residual matrices while preserving training stability.

The Non-Local Network (NLNet) presents a pioneering approach for capturing long-range dependencies within an image, via aggregating query-specific global context to each query position. However, through a rigorous empirical analysis, we have found that the global contexts modeled by the non-local network are almost the same for different query positions. In this paper, we take advantage of this finding to create a simplified network based on a query-independent formulation, which maintains the accuracy of NLNet but with significantly less computation. We further replace the one-layer transformation function of the non-local block by a two-layer bottleneck, which further reduces the parameter number considerably. The resulting network element, called the global context (GC) block, effectively models global context in a lightweight manner, allowing it to be applied at multiple layers of a backbone network to form a global context network (GCNet). Experiments show that GCNet generally outperforms NLNet on major benchmarks for various recognition tasks. The code and network configurations are available at https://github.com/xvjiarui/GCNet.