Fast, Robust and Non-convex Subspace Recovery

TL;DR

This paper introduces FMS, a fast, robust, non-convex algorithm for subspace recovery that effectively handles outliers and has proven convergence properties, outperforming existing methods in speed and accuracy.

Contribution

The paper proposes the FMS algorithm, a novel non-convex approach for robust subspace recovery with theoretical convergence guarantees and superior computational efficiency.

Findings

FMS converges to a stationary point and near the global minimum under certain data models.

FMS has globally bounded iteration complexity and local r-linear convergence.

Numerical experiments show FMS's speed and accuracy outperform existing methods.

Abstract

This work presents a fast and non-convex algorithm for robust subspace recovery. The data sets considered include inliers drawn around a low-dimensional subspace of a higher dimensional ambient space, and a possibly large portion of outliers that do not lie nearby this subspace. The proposed algorithm, which we refer to as Fast Median Subspace (FMS), is designed to robustly determine the underlying subspace of such data sets, while having lower computational complexity than existing methods. We prove convergence of the FMS iterates to a stationary point. Further, under a special model of data, FMS converges to a point which is near to the global minimum with overwhelming probability. Under this model, we show that the iteration complexity is globally bounded and locally $r$-linear. The latter theorem holds for any fixed fraction of outliers (less than 1) and any fixed positive distance between the limit point and the global minimum. Numerical experiments on synthetic and real data demonstrate its competitive speed and accuracy.