Dual Principal Component Pursuit: Probability Analysis and Efficient Algorithms

Abstract

Recent methods for learning a linear subspace from data corrupted by outliers are based on convex 1 and nuclear norm optimization and require the dimension of the subspace and the number of outliers to be sufficiently small (Xu et al., 2010). In sharp contrast, the recently proposed Dual Principal Component Pursuit (DPCP) method (Tsakiris and Vidal, 2015) can provably handle subspaces of high dimension by solving a non-convex 1 optimization problem on the sphere. However, its geometric analysis is based on quantities that are difficult to interpret and are not amenable to statistical analysis. In this paper we provide a refined geometric analysis and a new statistical analysis that show that DPCP can tolerate as many outliers as the square of the number of inliers, thus improving upon other provably correct robust PCA methods. We also propose a scalable Projected Sub-Gradient Method method (DPCP-PSGM) for solving the DPCP problem and show it admits linear convergence even though the underlying optimization problem is non-convex and nonsmooth. Experiments on road plane detection from 3D point cloud data demonstrate that DPCP-PSGM can be more efficient than the traditional RANSAC algorithm, which is one of the most popular methods for such computer vision applications.

Finding the Sparsest Vectors in a Subspace: Theory, Algorithms, and Applications

“…For general nonconvex problems, Späth and Watson [84] established the convergence of ALP to a critical point. For the DPCP problem, this proving technique is further utilized in [103,127] to show the convergence to a target solution starting from a spectral initialization. Again, the latter result is achieved mainly due to the underlying benign geometric structures of the problem that we discussed in Section IV-B.…”