Linearized Optimal Transport on Manifolds

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-07-09 DOI:10.1137/23m1564535

Clément Sarrazin, Bernhard Schmitzer

引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4970-5016, August 2024.
Abstract. Optimal transport is a geometrically intuitive, robust, and flexible metric for sample comparison in data analysis and machine learning. Its formal Riemannian structure allows for a local linearization via a tangent space approximation. This in turn leads to a reduction of computational complexity and simplifies combination with other methods that require a linear structure. Recently this approach has been extended to the unbalanced Hellinger–Kantorovich (HK) distance. In this article we further extend the framework in various ways, including measures on manifolds, the spherical HK distance, a study of the consistency of discretization via the barycentric projection, and the continuity properties of the logarithmic map for the HK distance.

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曲面上的线性化最优传输

SIAM 数学分析期刊》，第 56 卷第 4 期，第 4970-5016 页，2024 年 8 月。摘要最优传输是一种直观、稳健、灵活的几何度量，可用于数据分析和机器学习中的样本比较。其形式上的黎曼结构允许通过切线空间近似实现局部线性化。这反过来又降低了计算复杂度，简化了与其他需要线性结构的方法的结合。最近，这种方法被扩展到非平衡海灵格-康托洛维奇（HK）距离。在这篇文章中，我们从多方面进一步扩展了这一框架，包括流形上的度量、球形 HK 距离、通过重心投影进行离散化的一致性研究，以及 HK 距离对数映射的连续性特性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.