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Nonsmooth rank-one matrix factorization landscape

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We provide the first positive result on the nonsmooth optimization landscape of robust principal component analysis, to the best of our knowledge. It is the object of several conjectures and remains mostly uncharted territory. We identify a necessary and sufficient condition for the absence of spurious local minima in the rank-one case. Our proof exploits the subdifferential regularity of the objective function in order to eliminate the existence quantifier from the first-order optimality condition known as Fermat’s rule.

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The authors acknowledge support from DTRA grant 13-1-0021, DARPA grant Lagrange, NSF grant 2023032, and ONR grant N00014-21-1-2282. Data sharing not applicable to this article as no datasets were generated or analysed during the current study

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Correspondence to Cédric Josz.

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Josz, C., Lai, L. Nonsmooth rank-one matrix factorization landscape. Optim Lett 16, 1611–1631 (2022).

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