提出新型距离度量，连接计算几何与机器学习。
Bregman-Hausdorff divergence: strengthening the connections between computational geometry and machine learning
- 基于Bregman散度扩展霍夫曼距离，支持非对称度量空间。
- 在高维概率预测对比中展现高效性，适用于百维以上数据。
- 兼具理论创新与实用算法，适合研究几何与学习的交叉方向者。
本文有两个目标。技术层面，我们将霍夫曼距离从度量空间推广到具备非对称距离度量的空间，聚焦于包含KL散度（相对熵）的Bregman散度族。作为概念验证，我们利用新提出的Bregman-Hausdorff散度比较由相对熵损失训练的不同机器学习模型生成的两组概率预测。所提算法即使在数百维的大规模输入下也表现出惊人的效率。此外，本文还提供了一篇综述，概述了Bregman几何的基础知识及计算几何算法，重点介绍与该几何兼容且对机器学习相关的算法。
原文摘要 · Abstract (English)
The purpose of this paper is twofold. On a technical side, we propose an extension of the Hausdorff distance from metric spaces to spaces equipped with asymmetric distance measures. Specifically, we focus on the family of Bregman divergences, which includes the popular Kullback--Leibler divergence (also known as relative entropy). As a proof of concept, we use the resulting Bregman--Hausdorff divergence to compare two collections of probabilistic predictions produced by different machine learning models trained using the relative entropy loss. The algorithms we propose are surprisingly efficient even for large inputs with hundreds of dimensions. In addition to the introduction of this technical concept, we provide a survey. It outlines the basics of Bregman geometry, as well as computational geometry algorithms. We focus on algorithms that are compatible with this geometry and are relevant for machine learning.
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