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Finding a transformation between two unknown probability distributions from finite samples is crucial for modeling complex data distributions and performing tasks such as sample generation, domain adaptation and statistical inference. One…
We introduce the proximal optimal transport divergence, a novel discrepancy measure that interpolates between information divergences and optimal transport distances via an infimal convolution formulation. This divergence provides a…
The AC Optimal Power Flow (AC-OPF) is a key building block in many power system applications. It determines generator setpoints at minimal cost that meet the power demands while satisfying the underlying physical and operational…
Fast and accurate generation of molecular conformers is desired for downstream computational chemistry and drug discovery tasks. Currently, training and sampling state-of-the-art diffusion or flow-based models for conformer generation…
In this paper we study the problem of convergence and generalization error bound of stochastic momentum for deep learning from the perspective of regularization. To do so, we first interpret momentum as solving an $\ell_2$-regularized…
Despite their empirical success, the internal mechanism by which transformer models align tokens during language processing remains poorly understood. This paper provides a mechanistic and theoretical explanation of token alignment in LLMs.…
Optimal Transport (OT) distances such as Wasserstein have been used in several areas such as GANs and domain adaptation. OT, however, is very sensitive to outliers (samples with large noise) in the data since in its objective function,…
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…
Flow-matching models provide a powerful framework for various applications, offering efficient sampling and flexible probability path modeling. These models are characterized by flows with low curvature in learned generative trajectories,…
Conditional Flow Matching (CFM), a simulation-free method for training continuous normalizing flows, provides an efficient alternative to diffusion models for key tasks like image and video generation. The performance of CFM in solving…
Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these…
This paper considers the problem of unconstrained minimization of smooth convex functions having Lipschitz continuous gradients with known Lipschitz constant. We recently proposed an optimized gradient method (OGM) for this problem and…
Achieving a balance between image quality (precision) and diversity (recall) is a significant challenge in the domain of generative models. Current state-of-the-art models primarily rely on optimizing heuristics, such as the Fr\'echet…
The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization…
Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability…
Objective functions that optimize deep neural networks play a vital role in creating an enhanced feature representation of the input data. Although cross-entropy-based loss formulations have been extensively used in a variety of supervised…
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…
Modern stochastic optimization pipelines increasingly rely on learned generative models to represent uncertainty, while downstream decisions are evaluated almost entirely through Monte Carlo scenarios. This shifts the operational object of…
A key challenge in modern deep learning theory is to explain the remarkable success of gradient-based optimization methods when training large-scale, complex deep neural networks. Though linear convergence of such methods has been proved…
Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study…