Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS

2026-06-28

AI Summary

This work addresses the limitations of classical multidimensional scaling (MDS) under heavy-tailed distributions, where conventional distance metrics fail to accurately capture the discrepancy between empirical and true distributions. To overcome this, the authors propose Max-D-SW, an enhanced sliced Wasserstein distance that aggregates contributions across multiple orthogonal directions via joint optimization, replacing the standard single-direction approach. This method substantially improves the visualization performance of MDS on heavy-tailed data while preserving statistical tractability. Moreover, it reveals a non-monotonic relationship between sample complexity and MDS performance. Theoretical analysis demonstrates that Max-D-SW achieves sample complexity bounds comparable to those of the original Max-Sliced Wasserstein distance, offering a favorable balance between computational efficiency and representational capacity.

Abstract

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.

Problem

• Multidimensional Scaling
• Wasserstein distance
• heavy-tailed distributions
• pattern recognition
• empirical distribution

Innovation

• Max-D-SW
• Multidimensional Scaling
• Wasserstein distance
• orthonormal bases
• sample complexity