Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ordinary differential equations for regularized variational problems involving semi-discrete optimal transport

Adrien Cances, Luca Nenna, Daniyar Omarov, Brendan Pass

Abstract

We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed ordinary differential equations in the regularization parameter. The initial conditions for these equations, corresponding to solutions to completely regularized problems, are typically known explicitly. The ODE can then be solved to recover the solution for an arbitrary degree of regularization; we verify that the solution is continuous in the regularization parameter, implying that taking the limit of the trajectory yields the solution to the fully unregularized problem. We establish analogous results for a version of the problem when the non-optimal transport term is not scaled with the regularization parameter. We exploit our characterization to numerically solve several example problems using standard ODE methods; this strategy exhibits superior robustness to alternatives such as Newton's method, as arbitrary initializations are not required.

Ordinary differential equations for regularized variational problems involving semi-discrete optimal transport

Abstract

We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed ordinary differential equations in the regularization parameter. The initial conditions for these equations, corresponding to solutions to completely regularized problems, are typically known explicitly. The ODE can then be solved to recover the solution for an arbitrary degree of regularization; we verify that the solution is continuous in the regularization parameter, implying that taking the limit of the trajectory yields the solution to the fully unregularized problem. We establish analogous results for a version of the problem when the non-optimal transport term is not scaled with the regularization parameter. We exploit our characterization to numerically solve several example problems using standard ODE methods; this strategy exhibits superior robustness to alternatives such as Newton's method, as arbitrary initializations are not required.

Paper Structure

This paper contains 7 sections, 8 theorems, 49 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Problem Statement and Preliminary Results
  3. The Governing ODE
  4. Parameter-Independent Functional
  5. Computational Examples
  6. Problems in 1-d
  7. Problems in 2-d

Key Result

Lemma 2.3

Let $f : {\mathbb R}^N\to{\mathbb R}$ be differentiable and $m$-strongly convex, and let $g_t:{\mathbb R}^N\to{\mathbb R}$, $t\in(0,1)$, be convex, $\mathcal{C}^2$ differentiable Lipschitz maps with Lipschitz constants uniformly bounded by $L$. For $t\in(0,1)$, let $\psi(t)$ be the unique solution o

Figures (4)

  • Figure 1: Time evolution of Laguerre cells for Problem \ref{['prob3_ODE']} with $P = \tfrac{1}{2}$ and $N = 4$
  • Figure 2: Time evolution of Laguerre cells for Problem \ref{['prob1_ODE']} with nonuniform density \ref{['NonUnifDens']}
  • Figure 3: Time evolution of Laguerre cells with 8 random points and uniform measure
  • Figure 4: Time evolution of Laguerre cells with $12$ target points along the scaled parabola and a uniform measure, computed using the \ref{['prob1_ODE']} solver

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 9 more