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Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-Order Convergence

Author

Listed:
  • Yuetian Luo

    (Data Science Institute, University of Chicago, Chicago, Illinois 60637)

  • Wen Huang

    (School of Mathematical Sciences, Xiamen University, Xiamen 361005, China)

  • Xudong Li

    (School of Data Science, Fudan University, Shanghai 200433, China)

  • Anru Zhang

    (Department of Biostatistics & Bioinformatics, Duke University, Durham, North Carolina 27710)

Abstract

In this paper, we propose a recursive importance sketching algorithm for rank-constrained least squares optimization (RISRO). The key step of RISRO is recursive importance sketching, a new sketching framework based on deterministically designed recursive projections, and it significantly differs from the randomized sketching in the literature. Several existing algorithms in the literature can be reinterpreted under this new sketching framework, and RISRO offers clear advantages over them. RISRO is easy to implement and computationally efficient, and the core procedure in each iteration is to solve a dimension-reduced least squares problem. We establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. We also discover a deep connection of RISRO to the Riemannian Gauss–Newton algorithm on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO.

Suggested Citation

  • Yuetian Luo & Wen Huang & Xudong Li & Anru Zhang, 2024. "Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-Order Convergence," Operations Research, INFORMS, vol. 72(1), pages 237-256, January.
    • Handle: RePEc:inm:oropre:v:72:y:2024:i:1:p:237-256
    DOI: 10.1287/opre.2023.2445
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    References listed on IDEAS

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    1. Bamdev Mishra & Gilles Meyer & Silvère Bonnabel & Rodolphe Sepulchre, 2014. "Fixed-rank matrix factorizations and Riemannian low-rank optimization," Computational Statistics, Springer, vol. 29(3), pages 591-621, June.
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