Flows, Scaling, and Entropy Revisited: a Unified Perspective via Optimizing Joint Distributions

TL;DR

This paper presents a unified optimization framework for classical problems like optimal transport, matrix scaling, and balancing, connecting them through joint distribution optimization to simplify analysis and algorithms.

Contribution

It introduces a novel unified perspective on these problems via joint distribution optimization, unifying their formulation and analysis.

Findings

Unified framework simplifies problem analysis.

Connects diverse problems through joint distribution perspective.

Provides a common algorithmic approach.

Abstract

In this short expository note, we describe a unified algorithmic perspective on several classical problems which have traditionally been studied in different communities. This perspective views the main characters -- the problems of Optimal Transport, Minimum Mean Cycle, Matrix Scaling, and Matrix Balancing -- through the same lens of optimization problems over joint probability distributions P(x,y) with constrained marginals. While this is how Optimal Transport is typically introduced, this lens is markedly less conventional for the other three problems. This perspective leads to a simple and unified framework spanning problem formulation, algorithm development, and runtime analysis.