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- W3213493131 abstract "We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based analysis in terms of the Bregman divergence allows us to establish convergence of mirror descent -- with different choices of the mirror maps -- to a matrix that, among all global minimizers of the empirical risk, minimizes a quantity explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann entropy. In both cases, this characterization implies that mirror descent, a first-order algorithm minimizing the unregularized empirical risk, recovers low-rank matrices under the same set of assumptions that are sufficient to guarantee recovery for nuclear-norm minimization. When the sensing matrices are symmetric and commute, we show that gradient descent with full-rank factorized parametrization is a first-order approximation to mirror descent, in which case we obtain an explicit characterization of the implicit bias of gradient flow as a by-product." @default.
- W3213493131 created "2021-11-22" @default.
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- W3213493131 date "2021-05-28" @default.
- W3213493131 modified "2023-10-06" @default.
- W3213493131 title "Implicit Regularization in Matrix Sensing via Mirror Descent" @default.
- W3213493131 doi "https://doi.org/10.48550/arxiv.2105.13831" @default.
- W3213493131 hasPublicationYear "2021" @default.
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