Outlier Detection and Robust PCA Using a Convex Measure of Innovation

Abstract

This paper presents a provable and strong algorithm, termed Innovation Search (iSearch), to robust Principal Component Analysis (PCA) and outlier detection. An outlier by definition is a data point which does not participate in forming a low dimensional structure with a large number of data points in the data. In other word, an outlier carries some innovation with respect to most of the other data points. iSearch ranks the data points based on their values of innovation. A convex optimization problem is proposed whose optimal value is used as our measure of innovation. We derive analytical performance guarantees for the proposed robust PCA method under different models for the distribution of the outliers including randomly distributed outliers, clustered outliers, and linearly dependent outliers. Moreover, it is shown that iSearch provably recovers the span of the inliers when the inliers lie in a union of subspaces. In the challenging scenarios in which the outliers are close to each other or they are close to the span of the inliers, iSearch is shown to outperform most of the existing methods.