Revisiting L21-norm Robustness with Vector Outlier Regularization

TL;DR

This paper introduces the Vector Outlier Regularization (VOR) framework to analyze the robustness of the L2,1 norm, revealing its properties and proposing a new VORPCA model for matrix factorization that outperforms traditional methods.

Contribution

The paper proposes the VOR framework to understand L2,1 norm robustness and introduces VORPCA, a novel matrix factorization method leveraging this framework.

Findings

VOR unmask the independence of outlier effects from their distance.

L2,1 norm is a limiting case of VOR as the threshold shrinks.

VORPCA outperforms trace-norm based L21-norm PCA in experiments.

Abstract

In many real-world applications, data usually contain outliers. One popular approach is to use L2,1 norm function as a robust error/loss function. However, the robustness of L2,1 norm function is not well understood so far. In this paper, we propose a new Vector Outlier Regularization (VOR) framework to understand and analyze the robustness of L2,1 norm function. Our VOR function defines a data point to be outlier if it is outside a threshold with respect to a theoretical prediction, and regularize it-pull it back to the threshold line. We then prove that L2,1 function is the limiting case of this VOR with the usual least square/L2 error function as the threshold shrinks to zero. One interesting property of VOR is that how far an outlier lies away from its theoretically predicted value does not affect the final regularization and analysis results. This VOR property unmasks one of the most peculiar property of L2,1 norm function: The effects of outliers seem to be independent of how outlying they are-if an outlier is moved further away from the intrinsic manifold/subspace, the final analysis results do not change. VOR provides a new way to understand and analyze the robustness of L2,1 norm function. Applying VOR to matrix factorization leads to a new VORPCA model. We give a comprehensive comparison with trace-norm based L21-norm PCA to demonstrate the advantages of VORPCA.