Matrix Sensing with Kernel Optimal Loss: Robustness and Optimization Landscape

TL;DR

This paper investigates how kernel-based robust loss functions improve the robustness and optimization landscape in noisy matrix sensing, especially under non-Gaussian noise, through theoretical and empirical analysis.

Contribution

It introduces a kernel-based robust loss for matrix sensing that enhances robustness and reshapes the optimization landscape, with theoretical bounds and empirical validation.

Findings

Robust loss handles heavy-tailed noise effectively.

Optimization landscape is improved with fewer spurious minima.

Method outperforms traditional MSE loss in noisy settings.

Abstract

In this paper we study how the choice of loss functions of non-convex optimization problems affects their robustness and optimization landscape, through the study of noisy matrix sensing. In traditional regression tasks, mean squared error (MSE) loss is a common choice, but it can be unreliable for non-Gaussian or heavy-tailed noise. To address this issue, we adopt a robust loss based on nonparametric regression, which uses a kernel-based estimate of the residual density and maximizes the estimated log-likelihood. This robust formulation coincides with the MSE loss under Gaussian errors but remains stable under more general settings. We further examine how this robust loss reshapes the optimization landscape by analyzing the upper-bound of restricted isometry property (RIP) constants for spurious local minima to disappear. Through theoretical and empirical analysis, we show that this new loss excels at handling large noise and remains robust across diverse noise distributions. This work offers initial insights into enhancing the robustness of machine learning tasks through simply changing the loss, guided by an intuitive and broadly applicable analytical framework.