Matrix Sensing with Kernel Optimal Loss: Robustness and Optimization Landscape

2025-11-03

AI Summary

To address insufficient noise robustness in nonconvex matrix sensing, this paper identifies the vulnerability of conventional mean squared error (MSE) loss under non-Gaussian—particularly heavy-tailed—noise. We propose a nonparametric robust loss function based on kernel density estimation (KDE), which adaptively models arbitrary noise distributions by maximizing the likelihood of residuals. Under Gaussian assumptions, the loss reduces to MSE, preserving theoretical optimality while extending applicability. Theoretically, our loss weakens reliance on the restricted isometry property (RIP) and eliminates spurious local minima. Empirically, it significantly improves recovery accuracy and stability across diverse heavy-tailed noise settings, outperforming both MSE and state-of-the-art robust losses.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Adversarial Learning & Robustness
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Investigating loss function effects on robustness in noisy matrix sensing
• Developing kernel-based robust loss for non-Gaussian noise distributions
• Analyzing optimization landscape changes through RIP constant bounds

Innovation

• Kernel-based robust loss for non-Gaussian noise
• Optimization landscape analysis via RIP constants
• Nonparametric regression for residual density estimation