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Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that…
The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that…
Decision-focused learning (DFL) integrates predictive modeling and optimization by training predictors to optimize the downstream decision target rather than merely minimizing prediction error. To date, existing DFL methods typically rely…
Through Diffusion Models (DMs), we have made significant advances in generating high-quality images. Our exploration of these models delves deeply into their core operational principles by systematically investigating key aspects across…
We study entropy and optimal transport theory in the free probabilistic setting motivated by the large-$n$ theory of random tuples of matrices. We define a new version of free entropy $\chi_{\operatorname{chron}}^{\mathcal{U}}$, which is…
Diffusion models have emerged as powerful generative frameworks by progressively adding noise to data through a forward process and then reversing this process to generate realistic samples. While these models have achieved strong…
We study the problem of estimating a sequence of evolving probability distributions from historical data, where the underlying distribution changes over time in a nonstationary and nonparametric manner. To capture gradual changes, we…
This paper studies optical flow estimation, a critical task in motion analysis with applications in autonomous navigation, action recognition, and film production. Traditional optical flow methods require consecutive frames, which are often…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
Diffusion-based models on continuous spaces have seen substantial recent progress through the mathematical framework of gradient flows, leveraging the Wasserstein-2 (${W}_2$) metric via the Jordan-Kinderlehrer-Otto (JKO) scheme. Despite the…
The Diffusion Probabilistic Model (DPM) has demonstrated remarkable performance across a variety of generative tasks. The inherent randomness in diffusion models helps address issues such as blurring at the edges of medical images and…
Diffusion generative models are currently the most popular generative models. However, their underlying modeling process is quite complex, and starting directly with the seminal paper Denoising Diffusion Probability Model (DDPM) can be…
We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications. Existing…
Diffusion models (DMs) have become the dominant paradigm of generative modeling in a variety of domains by learning stochastic processes from noise to data. Recently, diffusion denoising bridge models (DDBMs), a new formulation of…
Enhancing the efficiency of high-quality image generation using Diffusion Models (DMs) is a significant challenge due to the iterative nature of the process. Flow Matching (FM) is emerging as a powerful generative modeling paradigm based on…
Generative modeling typically concerns transporting a single source distribution to a target distribution via simple probability flows. However, in fields like computer graphics and single-cell genomics, samples themselves can be viewed as…
We formulate a class of velocity-free finite-particle methods for mass transport problems based on a time-discrete incremental variational principle that combines entropy and the cost of particle transport, as measured by the Wasserstein…
We provide new convergence guarantees in Wasserstein distance for diffusion-based generative models, covering both stochastic (DDPM-like) and deterministic (DDIM-like) sampling methods. We introduce a simple framework to analyze…
Discrete flow models (DFMs) are a class of flexible generative models for generating discrete data, and diffusion large language models (dLLMs) can be viewed as a special case with a specific choice of mixture path and a masked source…
We analyze the gradient flow of a potential energy in the space of probability measures when we substitute the optimal transport geometry with a geometry based on Sinkhorn divergences, a debiased version of entropic optimal transport. This…