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The Wasserstein distance has emerged as a key metric to quantify distances between probability distributions, with applications in various fields, including machine learning, control theory, decision theory, and biological systems.…
In this paper we address the convergence of stochastic approximation when the functions to be minimized are not convex and nonsmooth. We show that the "mean-limit" approach to the convergence which leads, for smooth problems, to the ODE…
Out-of-distribution (OOD) detection is critical for ensuring the reliability of deep learning systems, particularly in safety-critical applications. Likelihood-based deep generative models have historically faced criticism for their…
This paper develops a rigorous probabilistic framework that extends denoising diffusion models to the setting of noncommutative random variables. Building on Voiculescu's theory of free entropy and free Fisher information, we formulate…
Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error…
We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of the Euler scheme of some diffusion process at a fixed deterministic…
We present a generative modeling framework for synthesizing physically feasible two-dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions. Whereas existing diffusion-based flow generators either ignore…
We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of…
We study conditional generation in diffusion models under hard constraints, where generated samples must satisfy prescribed events with probability one. Such constraints arise naturally in safety-critical applications and in rare-event…
The estimation of a log-concave density on $\mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet…
Score based approaches to sampling have shown much success as a generative algorithm to produce new samples from a target density given a pool of initial samples. In this work, we consider if we have no initial samples from the target…
We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density with respect to the Lebesgue measure on $\mathbb{R}^d$, known up to a normalization constant $x \mapsto \pi(x)=…
We propose an efficient framework for amortized conditional inference by leveraging exact conditional score-guided diffusion models to train a non-reversible neural network as a conditional generative model. Traditional normalizing flow…
This paper develops a generative model by minimizing the second-order Wasserstein loss (the $W_2$ loss) through a distribution-dependent ordinary differential equation (ODE), whose dynamics involves the Kantorovich potential associated with…
Fitting a function by using linear combinations of a large number $N$ of `simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to…
We study the proximal sampler of Lee, Shen, and Tian (2021) and obtain new convergence guarantees under weaker assumptions than strong log-concavity: namely, our results hold for (1) weakly log-concave targets, and (2) targets satisfying…
Diffusion models have demonstrated remarkable empirical success in the recent years and are considered one of the state-of-the-art generative models in modern AI. These models consist of a forward process, which gradually diffuses the data…
Diffusion models have made rapid progress in generating high-quality samples across various domains. However, a theoretical understanding of the Lipschitz continuity and second momentum properties of the diffusion process is still lacking.…
It has recently been shown that for compressive sensing, significantly fewer measurements may be required if the sparsity assumption is replaced by the assumption the unknown vector lies near the range of a suitably-chosen generative model.…
Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the…