Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS
2026-06-29 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explore a new way to improve a visualization method called Multidimensional Scaling (MDS) by changing how distances between data points are measured. They introduce Max-D-SW, which sums information over sets of directions instead of just one, and show it works better especially for complex data with heavy tails. They also prove that this new distance measure is still reliable when based on samples, and interestingly, having a more precise distance doesn’t always mean better visualization results in MDS. Their work helps understand how to pick distance measures for better pattern recognition visuals.
Multidimensional Scaling (MDS)Max-D-SWMax-Sliced Wasserstein DistanceHeavy-tailed distributionsOrthonormal basesDistance metricsSample complexityPattern recognitionVisualization
Authors
Flor Martinez-Sermeno, Arturo Jaramillo, Johan Van Horebeek
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.