Robust signal recovery in Hadamard 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 and a strong law of large numbers, generalizing a classical result of K.-T. Sturm. For the Fréchet 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 neither require the data nor the noise to be i.i.d., nor do we require the noise to be independent of the data. A resampling scheme is introduced to analyze the stability of the (non-symmetric) inductive mean with respect to data loss, data permutation, and noise, and sufficient conditions for its convergence are provided. These results suggest that means in Hadamard spaces are as robust as those 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.