19,052 research outputs found
Robust subspace recovery by Tyler's M-estimator
This paper considers the problem of robust subspace recovery: given a set of
points in , if many lie in a -dimensional subspace, then
can we recover the underlying subspace? We show that Tyler's M-estimator can be
used to recover the underlying subspace, if the percentage of the inliers is
larger than and the data points lie in general position. Empirically,
Tyler's M-estimator compares favorably with other convex subspace recovery
algorithms in both simulations and experiments on real data sets
Robust PCA by Manifold Optimization
Robust PCA is a widely used statistical procedure to recover a underlying
low-rank matrix with grossly corrupted observations. This work considers the
problem of robust PCA as a nonconvex optimization problem on the manifold of
low-rank matrices, and proposes two algorithms (for two versions of
retractions) based on manifold optimization. It is shown that, with a proper
designed initialization, the proposed algorithms are guaranteed to converge to
the underlying low-rank matrix linearly. Compared with a previous work based on
the Burer-Monterio decomposition of low-rank matrices, the proposed algorithms
reduce the dependence on the conditional number of the underlying low-rank
matrix theoretically. Simulations and real data examples confirm the
competitive performance of our method
Disentangling Orthogonal Matrices
Motivated by a certain molecular reconstruction methodology in cryo-electron
microscopy, we consider the problem of solving a linear system with two unknown
orthogonal matrices, which is a generalization of the well-known orthogonal
Procrustes problem. We propose an algorithm based on a semi-definite
programming (SDP) relaxation, and give a theoretical guarantee for its
performance. Both theoretically and empirically, the proposed algorithm
performs better than the na\"{i}ve approach of solving the linear system
directly without the orthogonal constraints. We also consider the
generalization to linear systems with more than two unknown orthogonal
matrices
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