Complete Dictionary Recovery over the Sphere

We consider the problem of recovering a complete (i.e., square and invertible) matrix mathbf A0, from mathbf Y ∈ mathbb Rn × p with mathbf Y = mathbf A0 mathbf X0, provided mathbf X0 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 A0 when mathbf X0 has O(n) nonzeros per column, under suitable probability model for mathbf X0. In contrast, prior results based on efficient algorithms provide recovery guarantees when mathbf X0 has only O(n1-δ) nonzeros per column for any constant δ ∈ (0, 1). 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. To show this apparently hard problem is tractable, we first provide a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no "spurious" local minima. This particular geometric structure allows us to design a Riemannian trust region algorithm over the sphere that provably converges to one local minimizer with an arbitrary initialization, despite the presence of saddle points. The geometric approach we develop here may also shed light on other problems arising from nonconvex recovery of structured signals.