We reintroduce an M-estimator that was implicitly discussed by Tyler in 1987, to robustly recover the underlying linear model from a data set contaminated by outliers. We prove that the objective function of this estimator is geodesically convex on the manifold of all positive definite matrices, and propose a fast algorithm that obtains its unique minimum. Besides, we prove that when inliers (i.e., points that are not outliers) are sampled from a subspace and the percentage of outliers is bounded by some number, then under some very weak assumptions this algorithm can recover the underlying subspace exactly. We also show that our algorithm compares favorably with other convex algorithms of robust PCA empirically.
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