Robust Optimization and Inference on Manifolds

TL;DR

This paper introduces a robust, scalable optimization and inference method on manifolds using geometric medians, addressing challenges in modern data science applications with theoretical guarantees and empirical validation.

Contribution

It develops a novel robust estimator on manifolds based on median-of-means, with theoretical properties and practical demonstrations in data science tasks.

Findings

Proves key properties of geometric medians on manifolds.

Establishes robustness and concentration bounds for the estimator.

Demonstrates effectiveness on simulated and real datasets.

Abstract

We propose a robust and scalable procedure for general optimization and inference problems on manifolds leveraging the classical idea of `median-of-means' estimation. This is motivated by ubiquitous examples and applications in modern data science in which a statistical learning problem can be cast as an optimization problem over manifolds. Being able to incorporate the underlying geometry for inference while addressing the need for robustness and scalability presents great challenges. We address these challenges by first proving a key lemma that characterizes some crucial properties of geometric medians on manifolds. In turn, this allows us to prove robustness and tighter concentration of our proposed final estimator in a subsequent theorem. This estimator aggregates a collection of subset estimators by taking their geometric median over the manifold. We illustrate bounds on this estimator via calculations in explicit examples. The robustness and scalability of the procedure is illustrated in numerical examples on both simulated and real data sets.