A Novel M-Estimator for Robust PCA

摘要

We study the basic problem of robust subspace recovery. That is,we assume a data set that some of its points are sampled arounda fixed subspace and the rest of them are spread in the wholeambient space, and we aim to recover the fixed underlyingsubspace. We first estimate a€?robust inverse sample covariancea€?by solving a convex minimization procedure; we then recover thesubspace by the bottom eigenvectors of this matrix (their numbercorrespond to the number of eigenvalues close to 0). Weguarantee exact subspace recovery under some conditions on theunderlying data. Furthermore, we propose a fast iterativealgorithm, which linearly converges to the matrix minimizing theconvex problem. We also quantify the effect of noise andregularization and discuss many other practical and theoreticalissues for improving the subspace recovery in various settings.When replacing the sum of terms in the convex energy function(that we minimize) with the sum of squares of terms, we obtainthat the new minimizer is a scaled version of the inverse samplecovariance (when exists). We thus interpret our minimizer andits subspace (spanned by its bottom eigenvectors) as robustversions of the empirical inverse covariance and the PCAsubspace respectively. We compare our method with many otheralgorithms for robust PCA on synthetic and real data sets anddemonstrate state-of-the-art speed and accuracy. color="gray">