提出基于学习度量的k均值新理论，统一证明多种距离方法的稳定性。
$K-$means with learned metrics
- 在度量和测度未知时，用测量格罗莫夫-豪斯多夫拓扑分析k均值连续性
- 证明了基于Isomap、扩散距离等的k均值方法具有统计一致性
- 适用于几何学习、图数据及长度空间近似等场景，适合度量学习研究者
我们研究度量测度空间中弗雷歇克均值（Fréchet k-means）在测度与距离均未知且需估计的情形。证明了k均值在测量格罗莫夫-豪斯多夫拓扑下具有连续性，并给出了其决定的Voronoi簇的稳定性结果。不假设k均值集合的唯一性，但在唯一时结论更强。该框架为证明广泛度量学习方法的一致性提供了统一途径。具体应用包括：(i) 流形上的Isomap与费马测地距离；(ii) 扩散距离；(iii) 基于学习基底度量计算的Wasserstein距离。此外，还拓展至非统计范式如(i) 首达通过渗流和(ii) 长度空间的离散逼近。
原文摘要 · Abstract (English)
We study the Fréchet $k-$means of a metric measure space when both the measure and the distance are unknown and have to be estimated. We prove a general result that states that the $k-$means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also prove a stability result for the Voronoi clusters they determine. We do not assume uniqueness of the set of $k-$means, but when it is unique, the results are stronger. This framework provides a unified approach to proving consistency for a wide range of metric learning procedures. As concrete applications, we obtain new consistency results for several important estimators that were previously unestablished, even when $k=1$. These include $k-$means based on: (i) Isomap and Fermat geodesic distances on manifolds, (ii) difussion distances, (iii) Wasserstein distances computed with respect to learned ground metrics. Finally, we consider applications beyond the statistical inference paradigm like (iv) first passage percolation and (v) discrete approximations of length spaces.
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