An inexact Bregman proximal point method and its acceleration version for unbalanced optimal transport

TL;DR

This paper tackles Unbalanced Optimal Transport (UOT) with KL-relaxation by introducing IBPUOT, an inexact Bregman proximal point method that approximates proximal steps via entropy-regularized subproblems solved by Scaling. It establishes convergence guarantees with a $\mathcal{O}(1/N)$ rate under standard assumptions, and augments the framework with AIBPUOT, an accelerated variant exploiting the triangle scaling property to attain faster rates under suitable conditions. The method leverages a moderately sized entropy parameter to balance accuracy and stability, while often producing sparse transport plans without numerical overflow or underflow. Numerical experiments demonstrate superior stability, sparsity, and convergence speed of IBPUOT and AIBPUOT compared to Scaling and MM, confirming practical benefits for UOT in computational biology, imaging, and deep learning.

Abstract

The Unbalanced Optimal Transport (UOT) problem plays increasingly important roles in computational biology, computational imaging and deep learning. Scaling algorithm is widely used to solve UOT due to its convenience and good convergence properties. However, this algorithm has lower accuracy for large regularization parameters, and due to stability issues, small regularization parameters can easily lead to numerical overflow. We address this challenge by developing an inexact Bregman proximal point method for solving UOT. This algorithm approximates the proximal operator using the Scaling algorithm at each iteration. The algorithm (1) converges to the true solution of UOT, (2) has theoretical guarantees and robust regularization parameter selection, (3) mitigates numerical stability issues, and (4) can achieve comparable computational complexity to the Scaling algorithm in specific practice. Building upon this, we develop an accelerated version of inexact Bregman proximal point method for solving UOT by using acceleration techniques of Bregman proximal point method and provide theoretical guarantees and experimental validation of convergence and acceleration.

Figures (6)

• Figure 1: marginal distribution
• Figure 2: Calculate the approximate exact solution
• Figure 3: Comparison of convergence rates of various algorithms
• Figure 4: Comparison of convergence rates of IBPUOT with different $\beta$
• Figure 5: The transport plan of IBPUOT and Scaling algorithm

Theorems & Definitions (24)

• Definition 2.1: $\nu$ -subdifferentialchu2023efficient
• Definition 2.2: Legendre functionxie2020fast
• Definition 2.3: Bregman distancexie2020fast
• Lemma 2.1: three points identityxie2020fast
• Lemma 2.2: seelemaire1995convergence
• Lemma 3.1: sufficient descent property
• proof
• Theorem 3.2
• proof
• Definition 4.1: triangle scaling propertyhanzely2021accelerated