What is a Sketch-and-Precondition Derivation for Low-Rank Approximation? Inverse Power Error or Inverse Power Estimation?

• URL: http://arxiv.org/abs/2502.07993v1
• Date: Tue, 11 Feb 2025 22:19:56 GMT
• Title: What is a Sketch-and-Precondition Derivation for Low-Rank Approximation? Inverse Power Error or Inverse Power Estimation?
• Authors: Ruihan Xu, Yiping Lu,
• Abstract summary: We develop a sketch-and-precondition framework for randomized algorithms in low- rank matrix approximation. Our method achieves theoretical guarantees, including a convergence rate that improves at least linearly with the sketch size.
• Score: 3.817486368952521
• License:
• Abstract: Randomized sketching accelerates large-scale numerical linear algebra by reducing computa- tional complexity. While the traditional sketch-and-solve approach reduces the problem size di- rectly through sketching, the sketch-and-precondition method leverages sketching to construct a computational friendly preconditioner. This preconditioner improves the convergence speed of iterative solvers applied to the original problem, maintaining accuracy in the full space. Further- more, the convergence rate of the solver improves at least linearly with the sketch size. Despite its potential, developing a sketch-and-precondition framework for randomized algorithms in low- rank matrix approximation remains an open challenge. We introduce the Error-Powered Sketched Inverse Iteration (EPSI) Method via run sketched Newton iteration for the Lagrange form as a sketch-and-precondition variant for randomized low-rank approximation. Our method achieves theoretical guarantees, including a convergence rate that improves at least linearly with the sketch size.

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