# Relative Wasserstein Angle and the Problem of the W2-Nearest Gaussian Distribution

arXiv: https://arxiv.org/abs/2601.22355 · HTML: https://mlsanity.com/paper/2601.22355/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 29 Jan 2026.

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Datasheets for Datasets](https://mlsanity.com/paper/1803.09010/) (2018)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Data and its (dis)contents: A survey of dataset development and use in machine learning research](https://mlsanity.com/paper/2012.05345/) (2020)
- [EMNIST: an extension of MNIST to handwritten letters](https://mlsanity.com/paper/1702.05373/) (2017)
- [Relative Translation Invariant Wasserstein Distance](https://mlsanity.com/paper/2409.02416/) (2024)

## Abstract (arXiv)

Understanding the distributional structure of high-dimensional datasets has become an important topic, yet direct visual characterization is difficult. In this work, we develop a geometric framework for characterizing the distributional structure of empirical datasets by quantifying their deviation from the Gaussian family under the geometry induced by optimal transport theory. Building on the cone structure of the relative translation invariant quadratic Wasserstein (RW2) space, we define two geometric quantities---the relative Wasserstein angle and the orthogonal projection distance---and show that they are well-defined because of the flat geometry of the filling cone between distributional rays. This formulation recasts the problem of measuring deviation from the Gaussian family as an orthogonal projection problem onto the Gaussian cone and reveals that the commonly used moment-matching Gaussian is, in general, not the W2-nearest Gaussian to a non-Gaussian distribution. In one dimension, we derive closed-form expressions for the proposed quantities and extend closed-form expressions to several other location--scale families, including uniform, Laplace, and logistic distributions. In higher dimensions, we develop a numerical approximation method for the proposed quantities based on empirical optimal transport and covariance-shape optimization. Our experimental results show the empirical convergence and stability of the proposed methods and reveal that the RW2 angle provides a robust and consistent measure of distributional non-Gaussianity. Moreover, these results provide empirical support for its potential use as an indicator of distributional heterogeneity.