L1-norm Error Function Robustness and Outlier Regularization

In many real-world applications, data come with corruptions, large errors or outliers. One popular approach is to use -norm function. However, the robustness of -norm function is not well understood so far. In this paper, we present a new outlier regularization framework to understand and analyze the robustness of -norm function. There are two main features for the proposed outlier regularization. (1) A key property of outlier regularization is that how far an outlier lies away from its theoretically predicted value does not affect the other important feature of outlier regularization is that it has an equivalent continuous representation that closely relates to function. This provides a new way to understand and analyze the robustness of function. We apply our outlier regularization framework to PCA and propose an outlier regularized PCA (ORPCA) model. Comparing to the trace-norm based robust PCA, ORPCA has sevsuppression. (2) It can retain small high rank com-