Related papers: Fixed-Point Neural Optimal Transport without Impli…

This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve…

One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and…

The traveling salesman problem is a fundamental combinatorial optimization problem with strong exact algorithms. However, as problems scale up, these exact algorithms fail to provide a solution in a reasonable time. To resolve this, current…

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to…

Motivated, in particular, by the entropy-regularized optimal transport problem, we consider convex optimization problems with linear equality constraints, where the dual objective has Lipschitz $p$-th order derivatives, and develop two…

This paper proposes a novel federated algorithm that leverages momentum-based variance reduction with adaptive learning to address non-convex settings across heterogeneous data. We intend to minimize communication and computation overhead,…

We propose a new framework for formulating optimal transport distances between Markov chains. Previously known formulations studied couplings between the entire joint distribution induced by the chains, and derived solutions via a reduction…

We consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth…

Optimal transport (OT) provides effective tools for comparing and mapping probability measures. We propose to leverage the flexibility of neural networks to learn an approximate optimal transport map. More precisely, we present a new and…

The fundamental theorem of classical optimal transport establishes strong duality and characterizes optimizers through a complementary slackness condition. Milestones such as Brenier's theorem and the Kantorovich-Rubinstein formula are…

The Sinkhorn operator has recently experienced a surge of popularity in computer vision and related fields. One major reason is its ease of integration into deep learning frameworks. To allow for an efficient training of respective neural…

Many important machine learning applications involve regularized nonconvex bi-level optimization. However, the existing gradient-based bi-level optimization algorithms cannot handle nonconvex or nonsmooth regularizers, and they suffer from…

We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second…

Iterative approximation methods using backpropagation enable the optimization of neural networks, but they remain computationally expensive, especially when used at scale. This paper presents an efficient alternative for optimizing neural…

Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…

In this paper, we introduce a generalized dynamical unbalanced optimal transport framework by incorporating limited control input and mass dissipation, addressing limitations in conventional optimal transport for control applications. We…

In this paper, we develop unrolled neural networks to solve constrained optimization problems, offering accelerated, learnable counterparts to dual ascent (DA) algorithms. Our framework, termed constrained dual unrolling (CDU), comprises…

Many biological systems are observed through heterogeneous modalities, requiring transport models that couple dynamics across spaces while allowing mass variation. To address this challenge, we introduce Unbalanced Synchronized Optimal…

Despite its important applications in Machine Learning, min-max optimization of nonconvex-nonconcave objectives remains elusive. Not only are there no known first-order methods converging even to approximate local min-max points, but the…

We present a framework for bi-level trajectory optimization in which a system's dynamics are encoded as the solution to a constrained optimization problem and smooth gradients of this lower-level problem are passed to an upper-level…