Towards a median signal detector through the total Bregman divergence and its robustness analysis

TL;DR

This paper introduces a new family of geometric signal detectors based on medians of the total Bregman divergence, demonstrating improved robustness and performance over traditional mean-based methods through numerical simulations.

Contribution

It proposes a novel median-based detector using the total Bregman divergence for Hermitian matrices, with analytical influence function computation and robustness analysis.

Findings

TBD medians outperform mean counterparts in robustness.

Numerical algorithms effectively compute the median detectors.

Robustness to outliers is analytically demonstrated.

Abstract

A novel family of geometric signal detectors are proposed through medians of the total Bregman divergence (TBD), which are shown advantageous over the conventional methods and their mean counterparts. By interpreting the observation data as Hermitian positive-definite matrices, their mean or median play an essential role in signal detection. As is difficult to be solved analytically, we propose numerical solutions through Riemannian gradient descent algorithms or fixed-point algorithms. Beside detection performance, robustness of a detector to outliers is also of vital importance, which can often be analyzed via the influence functions. Introducing an orthogonal basis for Hermitian matrices, we are able to compute the corresponding influence functions analytically and exactly by solving a linear system, which is transformed from the governing matrix equation. Numerical simulations show that the TBD medians are more robust than their mean counterparts.