Toric Geometry of Entropic Regularization

TL;DR

This paper explores the geometric interpretation of entropic regularization in large-scale linear programming, comparing it with log-barrier methods, and extends the framework to unbalanced optimal transport using toric varieties and optimal conic couplings.

Contribution

It introduces a geometric perspective on entropic regularization via toric varieties, compares it with existing methods, and develops new tools for unbalanced optimal transport.

Findings

Computed the degree of the associated toric variety.

Compared entropic regularization with log-barrier methods.

Explored algorithms like iterative scaling for these methods.

Abstract

Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport, and we develop the use of optimal conic couplings. We compute the degree of the associated toric variety, and we explore algorithms like iterative scaling.