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• W2601846890 abstract "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 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 $mathbf X_0$. In contrast, prior results based on efficient algorithms either only guarantee recovery when $mathbf X_0$ has $O(sqrt{n})$ zeros per column, or require multiple rounds of SDP relaxation to work when $mathbf X_0$ has $O(n^{1-delta})$ nonzeros per column (for any constant $delta in (0, 1)$). } 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." @default.
• W2601846890 created "2017-04-07" @default.
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• W2601846890 date "2017-02-01" @default.
• W2601846890 modified "2023-10-17" @default.
• W2601846890 title "Complete Dictionary Recovery Over the Sphere I: Overview and the Geometric Picture" @default.
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• W2601846890 doi "https://doi.org/10.1109/tit.2016.2632162" @default.
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