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A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrieval (2304.12522v2)
Published 25 Apr 2023 in math.OC, cs.LG, eess.SP, stat.CO, and stat.ML
Abstract: This paper considers the robust phase retrieval problem, which can be cast as a nonsmooth and nonconvex optimization problem. We propose a new inexact proximal linear algorithm with the subproblem being solved inexactly. Our contributions are two adaptive stopping criteria for the subproblem. The convergence behavior of the proposed methods is analyzed. Through experiments on both synthetic and real datasets, we demonstrate that our methods are much more efficient than existing methods, such as the original proximal linear algorithm and the subgradient method.
- J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature, vol. 400, no. 6742, pp. 342–344, 1999.
- R. P. Millane, “Phase retrieval in crystallography and optics,” Journal of the Optical Society of America A, vol. 7, no. 3, pp. 394–411, 1990.
- A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems, vol. 27, no. 1, p. 015005, 2010.
- C. Fienup and J. Dainty, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory and Application, vol. 231, p. 275, 1987.
- J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annual Review of Physical Chemistry, vol. 59, pp. 387–410, 2008.
- M. Fickus, D. G. Mixon, A. A. Nelson, and Y. Wang, “Phase retrieval from very few measurements,” Linear Algebra and its Applications, vol. 449, pp. 475–499, 2014.
- E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Transactions on Information Theory, vol. 61, no. 4, pp. 1985–2007, 2015.
- Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Communications on Pure and Applied Mathematics, vol. 70, no. 5, pp. 822–883, 2017.
- G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving systems of random quadratic equations via truncated amplitude flow,” IEEE Transactions on Information Theory, vol. 64, no. 2, pp. 773–794, 2017.
- H. Zhang, Y. Zhou, Y. Liang, and Y. Chi, “A nonconvex approach for phase retrieval: Reshaped wirtinger flow and incremental algorithms,” Journal of Machine Learning Research, vol. 18, 2017.
- J. C. Duchi and F. Ruan, “Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval,” Information and Inference: A Journal of the IMA, vol. 8, no. 3, pp. 471–529, 2019.
- H. Zhang, Y. Chi, and Y. Liang, “Provable non-convex phase retrieval with outliers: Median truncated wirtinger flow,” in International Conference on Machine Learning. PMLR, 2016, pp. 1022–1031.
- D. Davis, D. Drusvyatskiy, K. J. MacPhee, and C. Paquette, “Subgradient methods for sharp weakly convex functions,” Journal of Optimization Theory and Applications, vol. 179, no. 3, pp. 962–982, 2018.
- A. S. Lewis and S. J. Wright, “A proximal method for composite minimization,” Mathematical Programming, vol. 158, no. 1, pp. 501–546, 2016.
- D. Drusvyatskiy and C. Paquette, “Efficiency of minimizing compositions of convex functions and smooth maps,” Mathematical Programming, vol. 178, no. 1, pp. 503–558, 2019.
- J. C. Duchi and F. Ruan, “Stochastic methods for composite and weakly convex optimization problems,” SIAM Journal on Optimization, vol. 28, no. 4, pp. 3229–3259, 2018.
- V. Charisopoulos, Y. Chen, D. Davis, M. Díaz, L. Ding, and D. Drusvyatskiy, “Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence,” Foundations of Computational Mathematics, pp. 1–89, 2021.
- V. Charisopoulos, D. Davis, M. Díaz, and D. Drusvyatskiy, “Composite optimization for robust blind deconvolution,” arXiv preprint arXiv:1901.01624, 2019.
- Z. Wang, B. Liu, S. Chen, S. Ma, L. Xue, and H. Zhao, “A manifold proximal linear method for sparse spectral clustering with application to single-cell RNA sequencing data analysis,” INFORMS Journal on Optimization, vol. 4, no. 2, pp. 200–214, 2022.
- Y. Nesterov, “On an approach to the construction of optimal methods of minimization of smooth convex functions,” Ekonomika i Mateaticheskie Metody, vol. 24, no. 3, pp. 509–517, 1988.
- A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 183–202, 2009.
- Y. Nesterov, “Gradient methods for minimizing composite functions,” Mathematical Programming, vol. 140, no. 1, pp. 125–161, 2013.
- M. Díaz and B. Grimmer, “Optimal convergence rates for the proximal bundle method,” SIAM Journal on Optimization, vol. 33, no. 2, pp. 424–454, 2023.
- S. Bonettini, I. Loris, F. Porta, and M. Prato, “Variable metric inexact line-search-based methods for nonsmooth optimization,” SIAM Journal on Optimization, vol. 26, no. 2, pp. 891–921, 2016.
- C.-p. Lee and S. J. Wright, “Inexact successive quadratic approximation for regularized optimization,” Computational Optimization and Applications, vol. 72, no. 3, pp. 641–674, 2019.
- ——, “Inexact variable metric stochastic block-coordinate descent for regularized optimization,” Journal of Optimization Theory and Applications, vol. 185, no. 1, pp. 151–187, 2020.
- N. Parikh and S. Boyd, “Block splitting for distributed optimization,” Mathematical Programming Computation, vol. 6, no. 1, pp. 77–102, 2014.
- C. Dünner, S. Forte, M. Takác, and M. Jaggi, “Primal-dual rates and certificates,” in International Conference on Machine Learning. PMLR, 2016, pp. 783–792.