Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture
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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. In contrast, prior results based on efficient algorithms either only guarantee recovery when X_0 has O(√(n)) zeros per column, or require multiple rounds of SDP relaxation to work when X_0 has O(n^1-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. In this paper, we provide a geometric characterization of the objective landscape. In particular, we show that the problem is highly structured: with high probability, (1) there are no "spurious" local minimizers; and (2) around all saddle points the objective has a negative directional curvature. This distinctive structure makes the problem amenable to efficient optimization algorithms. In a companion paper (arXiv:1511.04777), we design a second-order trust-region algorithm over the sphere that provably converges to a local minimizer from arbitrary initializations, despite the presence of saddle points.
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