Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections

Hiroyuki Ochiai; Yoshiyuki Sekiguchi; Hayato Waki

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arXiv · 2401.16689

Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections

Hiroyuki Ochiai·Yoshiyuki Sekiguchi·Hayato Waki

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Abstract

We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to give the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at $O(k^{-1/2})$, independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence and does not use error bounds.

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Hiroyuki Ochiai, Yoshiyuki Sekiguchi, Hayato Waki. 2024-01-30. Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections. https://arxiv.org/abs/2401.16689

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