Dual Principal Component Pursuit
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We consider the problem of outlier rejection in single subspace learning. Classical approaches work with a direct representation of the subspace, and are thus efficient when the subspace dimension is small. Our approach works with a dual representation of the subspace and hence aims to find its orthogonal complement; as such, it is particularly suitable for subspaces whose dimension is very close to the ambient dimension (subspaces of high relative dimension). We pose the problem of computing normal vectors to the subspace as a non-convex ℓ_1 minimization problem on the sphere, which we call Dual Principal Component Pursuit (DPCP) problem. We provide theoretical guarantees, under which every global solution of DPCP is a vector in the orthogonal complement of the inlier subspace. Moreover, we relax the non-convex DPCP problem to a recursion of linear programming problems, which, as we show, converges in a finite number of steps to a vector orthogonal to the subspace. In particular, when the inlier subspace is a hyperplane, then the linear programming recursion converges in a finite number of steps to the global minimum of the non-convex DPCP problem. We also propose algorithms based on alternating minimization and Iteratively Reweighted Least-Squares, that are suitable for dealing with large-scale data. Extensive experiments on synthetic data show that the proposed methods are able to handle more outliers and higher relative dimensions than the state-of-the-art methods, while experiments with real face and object images show that our DPCP-based methods are competitive to the state-of-the-art.
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