异质空间中概率测度的快速鲁棒比较
Fast and Robust Comparison of Probability Measures in Heterogeneous Spaces
摘要：
在机器学习中，比较异构空间上支持的两个概率测度是一个越来越重要的问题。例如，当比较两个生物细胞群体时，每个生物细胞群体都有自己的一组特征，或者当观察在不同语料库/语言中训练的单词嵌入家族时，就会出现这样的问题。对于这样的设置，Gromov Wasserstein（GW）距离通常被视为黄金标准。GW是直观的，因为它量化了一个度量是否可以同构映射到另一个度量。然而，它的精确计算是棘手的，并且声称近似它的大多数算法仍然昂贵。\cite{memoli-2011}提出将每个分布中的每个点表示为其到所有其他点的距离的一维分布，在本文中，我们引入锚能量（AE）和锚Wasserstein（AW）距离，它们分别是在这种表示上实例化的能量和瓦瑟斯坦距离。我们的主要贡献是提出了一个扫描线算法来计算AE \emph{exactly}在对数二次时间，其中一个天真的实现将是立方。这是准线性w.r.t.问题本身的描述。我们的第二个贡献是建议的强大的变体AE和AW使用秩统计量，而不是原来的距离。我们表明，AE和AW在各种实验设置中表现良好，在一小部分的流行GW近似的计算成本。代码可在\url{https://github.com/joisino/anchor-energy}获得。
英文摘要：
Comparing two probability measures supported on heterogeneous spaces is an increasingly important problem in machine learning. Such problems arise when comparing for instance two populations of biological cells, each described with its own set of features, or when looking at families of word embeddings trained across different corpora/languages. For such settings, the Gromov Wasserstein (GW) distance is often presented as the gold standard. GW is intuitive, as it quantifies whether one measure can be isomorphically mapped to the other. However, its exact computation is intractable, and most algorithms that claim to approximate it remain expensive. Building on \cite{memoli-2011}, who proposed to represent each point in each distribution as the 1D distribution of its distances to all other points, we introduce in this paper the Anchor Energy (AE) and Anchor Wasserstein (AW) distances, which are respectively the energy and Wasserstein distances instantiated on such representations. Our main contribution is to propose a sweep line algorithm to compute AE \emph{exactly} in log-quadratic time, where a naive implementation would be cubic. This is quasi-linear w.r.t. the description of the problem itself. Our second contribution is the proposal of robust variants of AE and AW that uses rank statistics rather than the original distances. We show that AE and AW perform well in various experimental settings at a fraction of the computational cost of popular GW approximations. Code is available at \url{https://github.com/joisino/anchor-energy}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
