Outlier-Robust Orthogonal Regression on Manifolds

Tianjiao Ding, Liangzu Peng, Rene Vidal

Primary Area: optimization

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Keywords: Optimization over manifolds, orthogonal regression, subspace learning

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Abstract: Motivated by machine learning and computer vision applications, we formulate the problem of Outlier-Robust Orthogonal Regression to find a point in a manifold that satisfies as many linear equations as possible. Existing approaches addressing special cases of our formulation either lack theoretical support, are computationally costly, or somewhat ignore the manifold constraint; the latter two limit them from many applications. In this paper, we propose a unified approach based on solving a non-convex and non-smooth $\ell^1$ optimization problem over the manifold. We give conditions on the geometry of the input data, the manifold, and their interplay, under which the minimizers recover the ground truth; notably the conditions can hold even when the inliers are skewed within the true hyperplane. We provide a Riemannian subgradient method and an iteratively reweighted least squares method, suiting different computational oracles, and prove their linear/sub-linear convergence to minimizers/critical points. Experiments demonstrate that respecting the manifold constraints increases robustness against outliers in robust essential matrix estimation and robust rotation search.

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Submission Number: 6571