  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.

Machine Learning;Machine Learning