Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Evolution variational inequalities with general costs

Pierre-Cyril Aubin-Frankowski, Giacomo Enrico Sodini, Ulisse Stefanelli

TL;DR

The paper develops Evolution Variational Inequalities (EVIs) driven by general costs $c$ on general sets, extending gradient-flow theory beyond metric spaces. It establishes equivalences among differential, integral, and exponential-integral EVI formulations, proves contraction and energy identities, and provides local slope characterizations under compatibility conditions. Existence of EVI flows is shown as the limit of splitting schemes based on the $c$-transform, with both implicit and explicit iterations analyzed under cross-convexity and NNCC-type assumptions. The framework encompasses classical costs like $c=d^p$, KL divergence, and Sinkhorn divergences, enabling new gradient-flow-type evolutions in non-metric settings with entropic and Bregman structures. This yields a versatile toolbox for analyzing evolution in measures and other spaces where nonmetric costs govern the dynamics, with concrete schemes and convergence guarantees bridging discrete approximations to continuous EVI flows.

Abstract

We extend the theory of gradient flows beyond metric spaces by studying evolution variational inequalities (EVIs) driven by general cost functions $c$, including Bregman and entropic transport divergences. We establish several properties of the resulting flows, including stability and energy identities. Using novel notions of convexity related to costs $c$, we prove that EVI flows are the limit of splitting schemes, providing assumptions for both implicit and explicit iterations.

Evolution variational inequalities with general costs

TL;DR

The paper develops Evolution Variational Inequalities (EVIs) driven by general costs on general sets, extending gradient-flow theory beyond metric spaces. It establishes equivalences among differential, integral, and exponential-integral EVI formulations, proves contraction and energy identities, and provides local slope characterizations under compatibility conditions. Existence of EVI flows is shown as the limit of splitting schemes based on the -transform, with both implicit and explicit iterations analyzed under cross-convexity and NNCC-type assumptions. The framework encompasses classical costs like , KL divergence, and Sinkhorn divergences, enabling new gradient-flow-type evolutions in non-metric settings with entropic and Bregman structures. This yields a versatile toolbox for analyzing evolution in measures and other spaces where nonmetric costs govern the dynamics, with concrete schemes and convergence guarantees bridging discrete approximations to continuous EVI flows.

Abstract

We extend the theory of gradient flows beyond metric spaces by studying evolution variational inequalities (EVIs) driven by general cost functions , including Bregman and entropic transport divergences. We establish several properties of the resulting flows, including stability and energy identities. Using novel notions of convexity related to costs , we prove that EVI flows are the limit of splitting schemes, providing assumptions for both implicit and explicit iterations.
Paper Structure (18 sections, 21 theorems, 146 equations, 1 table)

This paper contains 18 sections, 21 theorems, 146 equations, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Main contributions
  4. Related work
  5. Plan of the paper
  6. Evolution Variational Inequalities with general cost c
  7. Basic notation
  8. Costs
  9. Conditions for compatibility of the topology s with c,p
  10. Equivalent formulations and main properties of the EVI
  11. On slopes and local characterizations
  12. On general costs and slopes
  13. A local characterization of $\mathop{\mathrm{\rm EVI}}\nolimits_\lambda(X,c,\phi)$
  14. Existence of EVI solutions as limit of splitting schemes
  15. Definition of the scheme through alternating minimization
  16. ...and 3 more sections

Key Result

Lemma 2.2

Let $c$ be a cost on $X$ satisfying eq:addass, let $\sigma$ be a Hausdorff topology on $X$ such that $c$ is jointly $\sigma$-lower semicontinuous, and such that the sublevels of $\phi:X\to(-\infty,+\infty]$ are $\sigma$-compact. Then, $\sigma$ is forward-Cauchy-compatible with $(c,\phi)$.

Theorems & Definitions (63)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 2.1: Compatible topology
  • Lemma 2.2: $\phi$ with $\sigma$-compact sublevels
  • proof
  • Lemma 2.3: $c$ regular semimetric
  • proof
  • Theorem 2.4: Equivalent EVI definitions
  • proof
  • ...and 53 more