Robust Signal Recovery in Hadamard Spaces

TL;DR

This paper investigates the stability and robustness of various means in Hadamard spaces, extending classical laws of large numbers and analyzing data contamination effects without requiring independence or identical distribution of data.

Contribution

It generalizes laws of large numbers for inductive and Fréchet means in Hadamard spaces, including robustness analysis under data contamination and non-i.i.d. conditions.

Findings

Inductive means satisfy a concentration inequality in quadratic mean.

Fréchet means obey a generalized law of large numbers in Hadamard spaces.

Means in Hadamard spaces are as robust as in Euclidean spaces.

Abstract

We analyze the stability of (strong) laws of large numbers in Hadamard spaces with respect to distributional perturbations. For the inductive means of a sequence of independent, but not necessarily identically distributed random variables, we provide a concentration inequality in quadratic mean, as well as a strong law of large numbers, generalizing a classical result of K.-T. Sturm. For the Fr\'echet mean, we generalize H. Ziezold's law of large numbers in Hadamard spaces. In this case, we neither require our data to be independent, nor identically distributed; reasonably mild conditions on the first two moments of our sample are enough. Additionally, we look at data contamination via a model inspired by Huber's $\varepsilon$-contamination model, in which we replace a random portion of the data with noise. In the most general setup, we do neither require the data, nor the noise to be i.i.d., nor do we require the noise to be independent of the data. To analyze the stability of the (non-symmetric) inductive mean with respect to data loss, data permutation, and noise, a resampling scheme is introduced, and sufficient conditions for its convergence are provided. These results suggest that means in Hadamard spaces are as robust as in Euclidean spaces. This is underlined by a small simulation study, in which we compare the robustness of means on the manifold of positive definite matrices, with means on open books.