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

arXiv: https://arxiv.org/abs/2606.29665 · HTML: https://mlsanity.com/paper/2606.29665/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 22 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices](https://mlsanity.com/paper/2411.10651/) (2024)

## Abstract (arXiv)

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.