arXiv subjects

Arthur Stéphanovitch

Publications and source records attributed to Arthur Stéphanovitch.

10 recordsLinked to original sources

arXiv2026

Convergence rates for generative drifting flows: fixed-scale obstructions and multihead acceleration

Drifting models offer a promising route to faster generative AI: they perform gradual transport during training, while generating new samples in a single step. This paper asks whether the underlying drifting process can converge rapidly to a target distribution under ideal conditions, before finite-data or optimization effects are introduced. We show that its convergence rate depends critically on how it handles spatial scale. With a single fixed resolution, fine-scale features of the target can become nearly invisible, leading to extremely slow convergence. We introduce a multihead approach that combines scale-normalized information across a continuum of resolutions. We prove that this multihead approach restores exponential convergence near standard reference distributions. These results identify fixed resolution as a key bottleneck and provide a simple route to faster one-step generative models.

cs.LG↗

arXiv2026

Statistical Analysis of Markovian Generative Modeling

These lecture notes introduce the statistical analysis of continuous-time generative models built from Markov dynamics. We begin with the stochastic-calculus foundations of score-based diffusion models, including time reversal, score matching, and sampling from learned scores. We then present the broader framework of generator matching, which describes flows, diffusions, jump processes, and discrete generative models through their infinitesimal generators. We then focus on finite-sample guarantees. We explain how errors in the learned drift or generator propagate to the final generated distribution, why stability and regularity properties are essential, and how time-adaptive neural network classes can achieve optimal Wasserstein rates for smooth target distributions. Overall, the notes aim to connect modern generative modeling algorithms with the probabilistic, analytic, and statistical tools needed to understand their worst-case performance.

math.ST↗

arXiv2026

Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities

Under general assumptions on the target distribution $p^\star$, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we obtain Wasserstein discretization bounds for Euler-type samplers in dimension $d$: with $N$ discretization steps, the error achieves the optimal rate $\sqrt{d}/N$ up to logarithmic factors. Moreover, the constants do not deteriorate exponentially with the spatial extent of $p^\star$. We also show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to $p^\star$, which implies Poincaré and log-Sobolev inequalities for a broad class of probability measures.

math.ST↗

arXiv2025

Generalization bounds for score-based generative models: a synthetic proof

We establish minimax convergence rates for score-based generative models (SGMs) under the $1$-Wasserstein distance. Assuming the target density $p^\star$ lies in a nonparametric $β$-smooth Hölder class with either compact support or subGaussian tails on $\mathbb{R}^d$, we prove that neural network-based score estimators trained via denoising score matching yield generative models achieving rate $n^{-(β+1)/(2β+d)}$ up to polylogarithmic factors. Our unified analysis handles arbitrary smoothness $β> 0$, supports both deterministic and stochastic samplers, and leverages shape constraints on $p^\star$ to induce regularity of the score. The resulting proofs are more concise, and grounded in generic stability of diffusions and standard approximation theory.

math.ST↗

arXiv2025

Regularity of the score function in generative models

We study the regularity of the score function in score-based generative models and show that it naturally adapts to the smoothness of the data distribution. Under minimal assumptions, we establish Lipschitz estimates that directly support convergence and stability analyses in both diffusion and ODE-based generative models. In addition, we derive higher-order regularity bounds, which simplify existing arguments for optimally approximating the score function using neural networks.

math.ST↗

arXiv2025

Generative model for optimal density estimation on unknown manifold

We propose a generative model that achieves minimax-optimal convergence rates for estimating probability distributions supported on unknown low-dimensional manifolds. Building on Fefferman's solution to the geometric Whitney problem, our estimator is itself supported on a submanifold that matches the regularity of the data's support. This geometric adaptation enables the estimator to be simultaneously minimax-optimal for all $ γ$-Hölder Integral Probability Metrics (IPMs) with $ γ\geq 1 $. We validate our approach through experiments on synthetic and real datasets, demonstrating competitive or superior performance compared to Wasserstein GAN and score-based generative models.

math.ST↗

arXiv2025

Smooth transport map via diffusion process

We extend the classical regularity theory of optimal transport to non-optimal transport maps generated by heat flow for perturbations of Gaussian measures. Considering probability measures of the form $dμ(x) = \exp\left(-\frac{|x|^2}{2} + a(x)\right)dx$ on $\mathbb{R}^d$ where $a$ has Hölder regularity $C^β$ with $β\geq 0$; we show that the Langevin map transporting the $d$-dimensional Gaussian distribution onto $μ$ achieves Hölder regularity $C^{β+ 1}$, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling.

math.PR↗

arXiv2025

Wasserstein GANs are Minimax Optimal Distribution Estimators

We provide non asymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure $μ$ with support in $\mathbb{R}^p$. The probability $μ$ is considered to be the push forward of the Lebesgue measure on the $d$-dimensional torus $\mathbb{T}^d$ by a map $g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$ of smoothness $β+1$. Measuring the error with the $γ$-Hölder Integral Probability Metric (IPM), we obtain up to logarithmic factors, the minimax optimal rate $O(n^{-\frac{β+γ}{2β+d}}\vee n^{-\frac{1}{2}})$ where $n$ is the sample size, $β$ determines the smoothness of the target measure $μ$, $γ$ is the smoothness of the IPM ($γ=1$ is the Wasserstein case) and $d\leq p$ is the intrinsic dimension of $μ$. In the process, we derive a sharp interpolation inequality between Hölder IPMs. This novel result of theory of functions spaces generalizes classical interpolation inequalities to the case where the measures involved have densities on different manifolds.

math.ST↗

arXiv2024

Integral Probability Metrics on submanifolds: interpolation inequalities and optimal inference

We study interpolation inequalities between Hölder Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $μ$ and $μ^\star$ have $β$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and $\mathcal{M}^\star$ respectively, then the Hölder IPMs $d_{\mathcal{H}^γ_1}$ of smoothness $γ\geq 1$ and $d_{\mathcal{H}^η_1}$ of smoothness $η>γ$, satisfy $d_{ \mathcal{H}_1^γ}(μ,μ^\star)\lesssim d_{ \mathcal{H}_1^η}(μ,μ^\star)^\frac{β+γ}{β+η}$, up to logarithmic factors. We provide an application of this result to high-dimensional inference. These functional inequalities turn out to be a key tool for density estimation on unknown submanifold. In particular, it allows to build the first estimator attaining optimal rates of estimation for all the distances $d_{\mathcal{H}_1^γ}$, $γ\in [1,\infty)$ simultaneously.

math.ST↗

arXiv2023

Optimal 1-Wasserstein Distance for WGANs

The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.

stat.ML↗