Complete Dictionary Recovery over the Sphere II: Recovery by Riemannian Trust-region Method
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We consider the problem of recovering a complete (i.e., square and invertible) matrix A_0, from Y ∈R^n × p with Y = A_0 X_0, provided X_0 is sufficiently sparse. This recovery problem is central to theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers A_0 when X_0 has O(n) nonzeros per column, under suitable probability model for X_0. Our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. In a companion paper (arXiv:1511.03607), we have showed that with high probability our nonconvex formulation has no "spurious" local minimizers and around any saddle point the objective function has a negative directional curvature. In this paper, we take advantage of the particular geometric structure, and describe a Riemannian trust region algorithm that provably converges to a local minimizer with from arbitrary initializations. Such minimizers give excellent approximations to rows of X_0. The rows are then recovered by linear programming rounding and deflation.
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