The Gaussian Transform

TL;DR

The Gaussian transform (GT) is a novel iterative method inspired by optimal transport for denoising and enhancing dataset structures, with theoretical stability guarantees and computational improvements over related methods.

Contribution

We introduce GT, establishing its stability and neighborhood properties, and improve its computational efficiency compared to mean shift and Wasserstein Transform methods.

Findings

GT is stable under perturbations.

GT neighborhoods are asymptotically ellipsoidal.

GT outperforms related methods in experiments.

Abstract

We introduce the Gaussian transform (GT), an optimal transport inspired iterative method for denoising and enhancing latent structures in datasets. Under the hood, GT generates a new distance function (GT distance) on a given dataset by computing the $\ell^2$-Wasserstein distance between certain Gaussian density estimates obtained by localizing the dataset to individual points. Our contribution is twofold: (1) theoretically, we establish firstly that GT is stable under perturbations and secondly that in the continuous case, each point possesses an asymptotically ellipsoidal neighborhood with respect to the GT distance; (2) computationally, we accelerate GT both by identifying a strategy for reducing the number of matrix square root computations inherent to the $\ell^2$-Wasserstein distance between Gaussian measures, and by avoiding redundant computations of GT distances between points via enhanced neighborhood mechanisms. We also observe that GT is both a generalization and a strengthening of the mean shift (MS) method, and it is also a computationally efficient specialization of the recently proposed Wasserstein Transform (WT) method. We perform extensive experimentation comparing their performance in different scenarios.