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• W3123272904 abstract "This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest $mathbf{x}^{natural}inmathbb{R}^{n}$ from $m$ quadratic equations/samples $y_{i}=(mathbf{a}_{i}^{top}mathbf{x}^{natural})^{2}$, $1leq ileq m$. This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning. We investigate the efficiency of gradient descent (or Wirtinger flow) designed for the nonconvex least squares problem. We prove that under Gaussian designs, gradient descent --- when randomly initialized --- yields an $epsilon$-accurate solution in $Obig(log n+log(1/epsilon)big)$ iterations given nearly minimal samples, thus achieving near-optimal computational and sample complexities at once. This provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval, without the need of (i) carefully-designed initialization, (ii) sample splitting, or (iii) sophisticated saddle-point escaping schemes. All of these are achieved by exploiting the statistical models in analyzing optimization algorithms, via a leave-one-out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data." @default.
• W3123272904 created "2021-02-01" @default.
• W3123272904 creator A5019973935 @default.
• W3123272904 creator A5031910872 @default.
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• W3123272904 date "2019-02-04" @default.
• W3123272904 modified "2023-10-16" @default.
• W3123272904 title "Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval" @default.
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• W3123272904 doi "https://doi.org/10.1007/s10107-019-01363-6" @default.
• W3123272904 hasPubMedCentralId "https://www.ncbi.nlm.nih.gov/pmc/articles/8025800" @default.
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• W3123272904 hasPublicationYear "2019" @default.
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