Low Rank Approximation in the Presence of Outliers

Authors Aditya Bhaskara, Srivatsan Kumar

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Aditya Bhaskara
  • School of Computing, University of Utah, Salt Lake City, UT, USA, http://www.cs.utah.edu/~bhaskara/
Srivatsan Kumar
  • School of Computing, University of Utah, Salt Lake City, UT, USA

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Aditya Bhaskara and Srivatsan Kumar. Low Rank Approximation in the Presence of Outliers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We consider the problem of principal component analysis (PCA) in the presence of outliers. Given a matrix A (d x n) and parameters k, m, the goal is to remove a set of at most m columns of A (outliers), so as to minimize the rank-k approximation error of the remaining matrix (inliers). While much of the work on this problem has focused on recovery of the rank-k subspace under assumptions on the inliers and outliers, we focus on the approximation problem. Our main result shows that sampling-based methods developed in the outlier-free case give non-trivial guarantees even in the presence of outliers. Using this insight, we develop a simple algorithm that has bi-criteria guarantees. Further, unlike similar formulations for clustering, we show that bi-criteria guarantees are unavoidable for the problem, under appropriate complexity assumptions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Low rank approximation
  • PCA
  • Robustness to outliers


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