GenAI4ELab/papercli-papers huggingface dataset:
Equivariant Manifold Neural ODEs and Differential Invariants
Authors: Emma Andersdotter; Daniel Persson; Fredrik Ohlsson
Venue: JMLR 2025 (main)

Abstract: In this paper we develop a geometric framework for equivariant manifold neural ordinary differential equations (NODEs), and use it to analyse their modelling capabilities for symmetric data. First, we consider the action of a Lie group $G$ on a smooth manifold $M$ and establish the equivalence between equivariance of vector fields, symmetries of the corresponding Cauchy problems, and equivariance of the associated NODEs. We also propose a novel formulation of the equivariant NODEs in terms of the differential invariants of the action of $G$ on $M$, based on Lie theory for symmetries of differential equations, which provides an efficient parameterisation of the space of equivariant vector fields in a way that is agnostic to both the manifold $M$ and the symmetry group $G$. Second, we construct augmented manifold NODEs through embeddings into equivariant flows, and show that they are universal approximators of equivariant diffeomorphisms on any connected $M$. Furthermore, we show that the augmented NODEs can be incorporated in the geometric framework and parametrised using higher order differential invariants. Finally, we consider the induced action of $G$ on different fields on $M$ and show how it generalises previous work, e.g., continuous normalizing flows, to equivariant models in any geometry.
The Z-Gromov-Wasserstein Distance
Authors: Martin Bauer; Facundo MÃ©moli; Tom Needham; Mao Nishino
Venue: JMLR 2025 (main)

Abstract: The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze data sets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.
O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
Authors: Gen Li; Yuling Yan
Venue: JMLR 2025 (main)

Abstract: Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for the denoising diffusion probabilistic model (DDPM), a widely used SDE-based sampler, under minimal assumptions. Our analysis shows that, provided $\ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. Moreover, we show that with careful coefficient design, the convergence rate improves to $O(k/T)$, where $k$ is the intrinsic dimension of the target data distribution. This highlights the ability of DDPM to automatically adapt to unknown low-dimensional structures, a common feature of natural image distributions. These results are achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.
Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity
Authors: Eliot Beyler; Francis Bach
Venue: JMLR 2025 (main)

Abstract: Score-based generative models achieve state-of-the-art sampling performance by denoising a distribution perturbed by Gaussian noise. In this paper, we focus on a single deterministic denoising step, and compare the optimal denoiser for the quadratic loss, we name "full-denoising", to the alternative "half-denoising" introduced by HyvÃ¤rinen (2025). We show that looking at the performance in terms of distance between distributions tells a more nuanced story, with different assumptions on the data leading to very different conclusions. We prove that half-denoising is better than full-denoising for regular enough densities, while full-denoising is better for singular densities such as mixtures of Dirac measures or densities supported on a low-dimensional subspace. In the latter case, we prove that full-denoising can alleviate the curse of dimensionality under a linear manifold hypothesis.
Zono-Conformal Prediction: Zonotope-Based Uncertainty Quantification for Regression and Classification Tasks
Authors: Laura LÃ¼tzow; Michael Eichelbeck; Mykel J. Kochenderfer; Matthias Althoff
Venue: JMLR 2025 (main)

Abstract: Conformal prediction is a popular uncertainty quantification method that augments a base predictor to return sets of predictions with statistically valid coverage guarantees. However, current methods are often computationally expensive and data-intensive, as they require constructing an uncertainty model before calibration. Moreover, existing approaches typically represent the prediction sets with intervals, which limits their ability to capture dependencies in multi-dimensional outputs. We address these limitations by introducing zono-conformal prediction, a novel approach inspired by interval predictor models and reachset-conformant identification that constructs prediction zonotopes with assured coverage. By placing zonotopic uncertainty sets directly into the model of the base predictor, zono-conformal predictors can be identified via a single, data-efficient linear program. While we can apply zono-conformal prediction to arbitrary nonlinear base predictors, we focus on feed-forward neural networks in this work. Aside from regression tasks, we also construct optimal zono-conformal predictors in classification settings where the output of an uncertain predictor is a set of possible classes. We provide probabilistic coverage guarantees and present methods for detecting outliers in the identification data. In extensive numerical experiments, we show that zono-conformal predictors are less conservative than interval predictor models and standard conformal prediction methods, while achieving a similar coverage over the test data.
Global Convergence of Adjoint-Optimized Neural PDEs
Authors: Konstantin Riedl; Justin Sirignano; Konstantinos Spiliopoulos
Venue: JMLR 2025 (main)

Abstract: Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from observed data in order to approximate missing or unresolved physics in the PDE model. The resulting neural-network PDE model, being a function of the neural network parameters, can be calibrated to the available ground truth data by optimizing over the PDE using gradient descent, where the gradient is evaluated in a computationally efficient manner by solving an adjoint PDE. These neural PDE models have emerged as an important research area in scientific machine learning. In this paper, we study the convergence of the adjoint gradient descent optimization method for training neural PDE models in the limit where both the number of hidden units and the training time tend to infinity. Specifically, for a general class of nonlinear parabolic PDEs with a neural network embedded in the source term, we prove convergence of the trained neural-network PDE solution to the target data (i.e., a global minimizer). The global convergence proof poses a unique mathematical challenge that is not encountered in finite-dimensional neural network convergence analyses due to (i) the neural network training dynamics involving a non-local neural network kernel operator in the infinite-width hidden layer limit where the kernel lacks a spectral gap for its eigenvalues and (ii) the nonlinearity of the limit PDE system, which leads to a non-convex optimization problem in the neural network function even in the infinite-width hidden layer limit (unlike in typical neural network training cases where the optimization problem becomes convex in the large neuron limit). The theoretical results are illustrated and empirically validated by numerical studies.
Kernel-based L_2-Boosting with Structure Constraints
Authors: Yao Wang; Xin Guo; Shao-Bo Lin
Venue: JMLR 2025 (main)

Abstract: Developing efficient kernel methods for regression is popular in the past two decades. In this paper, utilizing boosting on kernel-based weak learners, we propose a novel kernel-based learning algorithm called kernel-based re-scaled boosting with truncation, dubbed as KReBooT. The proposed KReBooT benefits in controlling the structure and producing sparse estimators, and is near overfitting resistant. We conduct both theoretical analysis and numerical simulations to illustrate the excellent performance of KReBooT. Theoretically, we prove that KReBooT can achieve the optimal numerical convergence rate for nonlinear approximation. Furthermore, using a variant of Talagrand's concentration inequality, we provide fast learning rates for KReBooT, which is a new record of boosting-type algorithms. Numerically, we carry out several simulations to show the promising performance of KReBooT in terms of its good generalization, near over-fitting resistance and structure constraints.
Geodesic Slice Sampling on the Sphere
Authors: Michael Habeck; Mareike Hasenpflug; Shantanu Kodgirwar; Daniel Rudolf
Venue: JMLR 2025 (main)

Abstract: Probability measures on the sphere form an important class of statistical models and are used, for example, in modeling directional data or shapes. Due to their widespread use, but also as an algorithmic building block, efficient sampling of distributions on the sphere is highly desirable. We propose a shrinkage based and an idealized geodesic slice sampling Markov chain, designed to generate approximate samples from distributions on the sphere. In particular, the shrinkage-based version of the algorithm can be implemented such that it runs efficiently and has no tuning parameters. We verify reversibility and prove that under weak regularity conditions geodesic slice sampling is uniformly ergodic. Numerical
experiments show that the proposed slice samplers achieve excellent mixing on challenging targets including distributions arising in rigid-registration problems and mixtures of von Mises-Fisher distributions. In these settings our approach outperforms standard samplers such as random-walk Metropolis-Hastings and Hamiltonian Monte Carlo.
A statistical perspective on algorithm unrolling models for inverse problems
Authors: Yves Atchade; Xinru Liu; Qiuyun Zhu
Venue: JMLR 2025 (main)

Abstract: We consider inverse problems where the forward model, that is the conditional distribution of the observation ${\bf y}\in\mathbb{R}^{d_y}$ given the latent variable of interest ${\bf x}\in\mathbb{R}^{d_x}$ is known, and access is given to a data set in which multiple instances of $({\bf x},{\bf y})$ are observed. In this context, algorithm unrolling has become a very popular approach for designing state-of-the-art deep neural network architectures that effectively exploit the forward model. We analyze the statistical properties of the gradient descent network (GDN), a well-known architecture driven by proximal gradient descent that epitomizes unrolling learning. Under some regularity conditions, we show that when $d_y\geq d_x$, the GDN estimator solves the inverse problem at a statistical rate faster than the nonparametric minimax rate achievable while ignoring the forward model. Furthermore, when the negative log-density of the latent variable ${\bf x}$ has a simple proximal operator, we show that GDN achieves the parametric rate $O(1/\sqrt{n})$. Furthermore, our results are explicit in the unrolling depth of the network and suggest that unrolling models are typically prone to overfitting as the unrolling depth increases, and careful tuning as function of the sample size is required for best performances. We provide several examples to illustrate these results.
Geometry-Dependent Matching Pursuit: a Transition Phase for Convergence on Linear Regression and LASSO
Authors: Celine Moucer; Adrien B. Taylor; Francis Bach
Venue: JMLR 2025 (main)

Abstract: Greedy first-order methods, such as coordinate descent with Gauss-Southwell rule or matching pursuit, have become popular in optimization due to their natural tendency to propose sparse solutions and their refined convergence guarantees. In this work, we propose a principled approach to generating (regularized) matching pursuit algorithms adapted to the geometry of the problem at hand, as well as their convergence guarantees. Building on these results, we derive approximate convergence guarantees and describe a transition phenomenon in the convergence of (regularized) matching pursuit from underparametrized to overparametrized models.
Continuum Attention for Neural Operators
Authors: Edoardo Calvello; Nikola B. Kovachki; Matthew E. Levine; Andrew M. Stuart
Venue: JMLR 2025 (main)

Abstract: 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.
Bayesian Score Calibration for Approximate Models
Authors: Joshua J. Bon; David J. Warne; David J. Nott; Christopher Drovandi
Venue: JMLR 2025 (main)

Abstract: Scientists continue to develop increasingly complex mechanistic models to reflect their knowledge more realistically. Statistical inference using these models can be challenging since the corresponding likelihood function is often intractable and model simulation may be computationally burdensome. Fortunately, in many of these situations it is possible to adopt a surrogate model or approximate likelihood function. It may be convenient to conduct Bayesian inference directly with a surrogate, but this can result in a posterior with poor uncertainty quantification. In this paper, we propose a new method for adjusting approximate posterior samples to reduce bias and improve posterior coverage properties. We do this by optimizing a transformation of the approximate posterior, the result of which maximizes a scoring rule. Our approach requires only a (fixed) small number of complex model simulations and is numerically stable. We develop supporting theory for our method and demonstrate beneficial corrections to approximate posteriors across several examples of increasing complexity.
Reward-Directed Score-Based Diffusion Models via q-Learning
Authors: Xuefeng Gao; Jiale Zha; Xun Yu Zhou
Venue: JMLR 2025 (main)

Abstract: We propose a new reinforcement learning (RL) formulation for training continuous-time score-based diffusion models for generative AI to generate samples that maximize reward functions while keeping the generated distributions close to the unknown target data distributions. Different from most existing studies, ours does not involve any pretrained model for the unknown score functions of the noise-perturbed data distributions, nor does it attempt to learn the score functions. Instead, we formulate the problem as entropy-regularized continuous-time RL and show that the optimal stochastic policy has a Gaussian distribution with a known covariance matrix.
Based on this result, we parameterize the mean of Gaussian policies and develop an actor--critic type (little) $q$-learning algorithm to solve the RL problem. A key ingredient in our algorithm design is to obtain noisy observations from the unknown score function via a ratio estimator. Our formulation can also be adapted to solve pure score-matching and fine-tuning pretrained models.
Numerically, we show the effectiveness of our approach by comparing its performance with two state-of-the-art RL methods that fine-tune pretrained models on several generative tasks including high-dimensional image generations. Finally, we discuss extensions of our RL formulation to probability flow ODE implementation of diffusion models and to conditional diffusion models.
Neural Operators Can Play Dynamic Stackelberg Games
Authors: Guillermo A. Alvarez; Ibrahim Ekren; Anastasis Kratsios; Xuwei Yang
Venue: JMLR 2025 (main)

Abstract: Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the follower's best-response operator can be approximately implemented by an attention-based neural operator, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.
Mixing Times and Privacy Analysis for the Projected Langevin Algorithm under a Modulus of Continuity
Authors: Mario Bravo; Juan P. Flores-Mella; CristÃ³bal GuzmÃ¡n
Venue: JMLR 2025 (main)

Abstract: We study the mixing time of the projected Langevin algorithm and the privacy curve of noisy stochastic gradient descent, beyond nonexpansive iterations. Specifically, we derive new mixing time bounds for the projected Langevin algorithm which are, in some important cases, dimension-free and poly-logarithmic in the accuracy, closely matching the existing results in the smooth convex case. Additionally, we establish new upper bounds for the privacy curve of the subsampled noisy stochastic gradient descent algorithm. These bounds show a crucial dependency on the regularity of gradients, and are useful for a wide range of convex losses beyond the smooth case. Our analysis relies on a suitable extension of the privacy amplification by iteration framework to noisy iterations whose gradient map is not necessarily nonexpansive. This extension is achieved by designing an optimization problem which accounts for the best possible RÃ©nyi divergence bound obtained by an application of privacy amplification by iteration, where the tractability of the problem is crucially related to the modulus of continuity of the associated gradient mapping. We show that, in several interesting cases, namely the nonsmooth convex, weakly smooth, and strongly dissipative cases, such an optimization problem can be solved exactly and explicitly, yielding the tightest possible privacy amplification by iteration based bounds.
Gradient Equilibrium in Online Learning: Theory and Applications
Authors: Anastasios N. Angelopoulos; Michael I. Jordan; Ryan J. Tibshirani
Venue: JMLR 2025 (main)

Abstract: We present a new perspective on online learning that we refer to as gradient equilibrium: a sequence of iterates achieves gradient equilibrium if the average of gradients of losses along the sequence converges to zero. In general, this condition is not implied by, nor implies, sublinear regret. It turns out that gradient equilibrium is achievable by standard online learning methods such as gradient descent and mirror descent with constant step sizes (rather than decaying step sizes, as is usually required for no regret). Further, as we show through examples, gradient equilibrium translates into an interpretable and meaningful property in online prediction problems spanning regression, classification, quantile estimation, and others. Notably, we show that the gradient equilibrium framework can be used to develop a debiasing scheme for black-box predictions under arbitrary distribution shift, based on simple post hoc online descent updates. We also show that post hoc gradient updates can be used to calibrate predicted quantiles under distribution shift, and that the framework leads to unbiased Elo scores for pairwise preference prediction.