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In many real-world applications, data come with corruptions, large errors oroutliers. One popular approach is to use L1-norm function. However, therobustness of L1-norm function is not well understood so far. In this paper, wepresent a new outlier regularization framework to understand and analyze therobustness of L1-norm function. There are two main features for the proposedoutlier regularization. (1) A key property of outlier regularization is thathow far an outlier lies away from its theoretically predicted value does notaffect the final regularization and analysis results. (2) Another importantfeature of outlier regularization is that it has an equivalent continuousrepresentation that closely relates to L1 function. This provides a new way tounderstand and analyze the robustness of L1 function. We apply our outlierregularization framework to PCA and propose an outlier regularized PCA (ORPCA)model. Comparing to the trace-normbased robust PCA, ORPCA has several benefits:(1) It does not suffer singular value suppression. (2) It can retain small highrank components which help retain fine details of data. (3) ORPCA can becomputed more efficiently.

L1-norm Error Function Robustness and Outlier Regularization