  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.
