改进MDS的度量方法，提升重尾分布下的可视化效果
Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS
- 用正交基聚合替代单一方向，优化Max-Sliced Wasserstein距离
- 在重尾分布下显著提升MDS可视化效果，数值表现更优
- 适用于高维数据模式识别，尤其适合处理复杂分布数据
本文研究了对多维尺度分析（MDS）中度量进行调整，以增强其在模式识别中的可视化能力。所提出的度量称为Max-D-SW，是最大切片Wasserstein距离的改进版本。与原始方法仅优化单个单位方向不同，Max-D-SW通过正交基聚合贡献，显著提升了在重尾分布下的MDS结果。我们还建立了样本复杂度界，表明Max-D-SW具有可比于其切片对应物的统计可处理性。此外，研究发现：尽管某度量具有更优的样本复杂度，但将其用于MDS输入时，并不必然带来更好性能。
原文摘要 · Abstract (English)
This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.
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