## A Novel M-Estimator for Robust PCA

*Teng Zhang, Gilad Lerman*; 15(Feb):749−808, 2014.

### Abstract

We study the basic problem of robust subspace recovery. That is,
we assume a data set that some of its points are sampled around
a fixed subspace and the rest of them are spread in the whole
ambient space, and we aim to recover the fixed underlying
subspace. We first estimate “robust inverse sample covariance”
by solving a convex minimization procedure; we then recover the
subspace by the bottom eigenvectors of this matrix (their number
correspond to the number of eigenvalues close to 0). We
guarantee exact subspace recovery under some conditions on the
underlying data. Furthermore, we propose a fast iterative
algorithm, which linearly converges to the matrix minimizing the
convex problem. We also quantify the effect of noise and
regularization and discuss many other practical and theoretical
issues 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 obtain
that the new minimizer is a scaled version of the inverse sample
covariance (when exists). We thus interpret our minimizer and
its subspace (spanned by its bottom eigenvectors) as robust
versions of the empirical inverse covariance and the PCA
subspace respectively. We compare our method with many other
algorithms for robust PCA on synthetic and real data sets and
demonstrate state-of-the-art speed and accuracy.

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