Geometric median and robust estimation in Banach spaces

TL;DR

This paper introduces a robust estimation method in Banach spaces that leverages the geometric median to achieve tight concentration bounds, effectively handling noisy and contaminated data in applications like regression and matrix recovery.

Contribution

It proposes a general framework for robust estimation in Banach spaces using the geometric median, improving deviation bounds under heavy-tailed noise.

Findings

The method provides tighter concentration bounds than individual estimators.

Applications include sparse linear regression and low-rank matrix recovery.

The approach is effective against heavy-tailed noise and outliers.

Abstract

In many real-world applications, collected data are contaminated by noise with heavy-tailed distribution and might contain outliers of large magnitude. In this situation, it is necessary to apply methods which produce reliable outcomes even if the input contains corrupted measurements. We describe a general method which allows one to obtain estimators with tight concentration around the true parameter of interest taking values in a Banach space. Suggested construction relies on the fact that the geometric median of a collection of independent "weakly concentrated" estimators satisfies a much stronger deviation bound than each individual element in the collection. Our approach is illustrated through several examples, including sparse linear regression and low-rank matrix recovery problems.