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 $\mathbf A_0$, from $\mathbf Y \in \mathbb{R}^{n \times p}$ with $\mathbf Y = \mathbf A_0 \mathbf X_0$, provided $\mathbf X_0$ is sufficiently sparse. This recovery problem is central to the 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 $\mathbf A_0$ when $\mathbf X_0$ has $O(n)$ nonzeros per column, under suitable probability model for $\mathbf X_0$. 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 any saddle point present is second-order. In this paper, we take advantage of the particular geometric structure and design a Riemannian trust region algorithm over the sphere that provably converges to a local minimizer with an arbitrary initialization. Such minimizers give excellent approximations to rows of $\mathbf X_0$. The rows are recovered by linear programming rounding and deflation.