# Testing and Learning on Distributions with Symmetric Noise Invariance

## TL;DR

This paper introduces noise-invariant distribution distances and features using kernel embeddings and MMD, enabling robust two-sample testing and learning despite symmetric additive noise.

## Contribution

It proposes new noise-invariant distribution distances and features, enhancing robustness in testing and learning on distributions with symmetric noise.

## Key findings

- Distances are invariant to symmetric additive noise
- Invariant features improve robustness in distribution learning
- Applicable to nonparametric two-sample testing

## Abstract

Kernel embeddings of distributions and the Maximum Mean Discrepancy (MMD), the resulting distance between distributions, are useful tools for fully nonparametric two-sample testing and learning on distributions. However, it is rarely that all possible differences between samples are of interest -- discovered differences can be due to different types of measurement noise, data collection artefacts or other irrelevant sources of variability. We propose distances between distributions which encode invariance to additive symmetric noise, aimed at testing whether the assumed true underlying processes differ. Moreover, we construct invariant features of distributions, leading to learning algorithms robust to the impairment of the input distributions with symmetric additive noise.