Complete Dictionary Recovery Over the Sphere II: Recovery by Riemannian Trust-Region Method

We consider the problem of recovering a complete (i.e., square and invertible) matrix A₀, from Y ∈ ℝ^{n × p} with Y = A₀ X₀, provided X₀ 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 A₀ when X₀ has O (n) nonzeros per column, under suitable probability model for X₀. 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, 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 the rows of X₀. The rows are then recovered by a linear programming rounding and deflation.