论文标题
在无与伦比的空间上为最佳运输的贡献
A contribution to Optimal Transport on incomparable spaces
论文作者
论文摘要
最佳传输是一种理论，允许定义概率分布之间距离的几何概念，并在点集之间找到对应关系，关系。许多机器学习应用程序来自该理论，在数学和优化之间的边界。本文提出了研究不同数据属于无与伦比空间的复杂情况。特别是我们解决以下问题：如何在图表之间定义和应用最佳传输，结构化数据之间？当数据变化并且不嵌入在同一度量空间中时，如何对其进行调整？本文为这些不同情况提出了一组最佳运输工具。一个重要的部分特别致力于研究Gromov-Wasserstein的距离，该距离允许在无与伦比的空间上定义有趣的运输问题。更广泛地，我们分析了各种提出的工具的数学特性，我们建立了算法解决方案来计算它们，并研究了它们在许多机器学习方案中的适用性，尤其涵盖了分类，简化，对结构化数据的分配，以及异质域的适应。
Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from this theory, at the frontier between mathematics and optimization. This thesis proposes to study the complex scenario in which the different data belong to incomparable spaces. In particular we address the following questions: how to define and apply Optimal Transport between graphs, between structured data? How can it be adapted when the data are varied and not embedded in the same metric space? This thesis proposes a set of Optimal Transport tools for these different cases. An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces. More broadly, we analyze the mathematical properties of the various proposed tools, we establish algorithmic solutions to compute them and we study their applicability in numerous machine learning scenarii which cover, in particular, classification, simplification, partitioning of structured data, as well as heterogeneous domain adaptation.
