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<p>We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to determine 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11590_2025_2196_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(k^{-\frac{1}{2}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, 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 without using error bounds.</p>

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Expansions of the characteristic polynomial of a perturbed PSD matrix and convergence analysis of alternating projections for the PSD cone and a line

  • 期刊论文

  • 发表日期：

    2025年4月9日

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  • Yoshiyuki Sekiguchi,
  • Hiroyuki Ochiai,
  • Hayato Waki

摘要

We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to determine 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 \(O(k^{-\frac{1}{2}})\) 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 without using error bounds.