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Analytic Formulas for Alternating Projection Sequences for the Positive Semidefinite Cone and an Application to Convergence Analysis

Hiroyuki Ochiai, Yoshiyuki Sekiguchi, Hayato Waki

TL;DR

This paper addresses the convergence of the alternating projection method between the affine subspace $E$ and the positive semidefinite cone $\mathbb{S}^n_+$. It introduces three analytic formulas that provide recursive and parametric descriptions of the AP sequence, including an eigenvalue-based formula for general cases, a second formula for the $\mathbb{S}^3_+$ rank-1 projection scenario, and rational formulas along a specially constructed slowest-converging curve. The authors show that the commonly cited singularity-degree upper bound can be tight in a singleton intersection of $\mathbb{S}^3_+$ with a $3$-plane, deriving convergence rates such as $O(k^{-1/6})$ along the slowest curve and linear rates in some cases. These results enable precise, computation-friendly convergence analysis beyond generic projection inequalities and provide a structured framework for understanding AP dynamics in low dimensions.

Abstract

We derive analytic formulas for the alternating projection method applied to the cone $\mathbb{S}^n_+$ of positive semidefinite matrices and an affine subspace. More precisely, we find recursive relations on parameters representing a sequence constructed by the alternating projection method. By applying these formulas, we analyze the alternating projection method in detail and show that the upper bound given by the singularity degree is actually tight when the alternating projection method is applied to $\mathbb{S}^3_+$ and a $3$-plane whose intersection is a singleton with singularity degree $2$.

Analytic Formulas for Alternating Projection Sequences for the Positive Semidefinite Cone and an Application to Convergence Analysis

TL;DR

This paper addresses the convergence of the alternating projection method between the affine subspace and the positive semidefinite cone . It introduces three analytic formulas that provide recursive and parametric descriptions of the AP sequence, including an eigenvalue-based formula for general cases, a second formula for the rank-1 projection scenario, and rational formulas along a specially constructed slowest-converging curve. The authors show that the commonly cited singularity-degree upper bound can be tight in a singleton intersection of with a -plane, deriving convergence rates such as along the slowest curve and linear rates in some cases. These results enable precise, computation-friendly convergence analysis beyond generic projection inequalities and provide a structured framework for understanding AP dynamics in low dimensions.

Abstract

We derive analytic formulas for the alternating projection method applied to the cone of positive semidefinite matrices and an affine subspace. More precisely, we find recursive relations on parameters representing a sequence constructed by the alternating projection method. By applying these formulas, we analyze the alternating projection method in detail and show that the upper bound given by the singularity degree is actually tight when the alternating projection method is applied to and a -plane whose intersection is a singleton with singularity degree .
Paper Structure (21 sections, 16 theorems, 143 equations, 5 figures)

This paper contains 21 sections, 16 theorems, 143 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The alternating projection method
  3. Contributions
  4. Organization
  5. Preliminaries
  6. Analytic formula for a general case
  7. Eigenvalue formula
  8. Applications of the eigenvalue formula
  9. Analytic formula when $P_{\mathbb{S}^3_+}(U)$ is rank $1$
  10. Analytic formula with a distance function
  11. Equations for the slowest curve
  12. Family of $3$-planes intersecting with $\mathbb{S}^3_+$ at a single point
  13. Parametrization
  14. Plücker embedding
  15. Rational formulas for a special curve
  16. ...and 6 more sections

Key Result

Proposition 3.1

Let $\tilde{p} = \varphi^{-1}\circ P_E\circ P_{\mathbb{S}^n_+} \circ \varphi(p)$ and $\lambda_1(p), \ldots, \lambda_n(p)$ be eigenvalues of $\varphi(p)$. Then we have where $n(p) = \{\ell \in [n]:\lambda_\ell(p) < 0\}$.

Figures (5)

  • Figure 1: The left figure displays a plot of $\|U_k - U_*\|$ with $t_0>0$ and the right figure displays a plot with $t_0<0$ in Example \ref{['ex:half']}, and the line fitting for the plot in the right figure.
  • Figure 2: Plot of $1/\|U_k-U_*\|^2$ with $t_0 > 0$ in Example 3.2 and the line fitting.
  • Figure 3: The left figure displays a log-plot of $\|U_k-U_*\|$ with $t_0 > 1$ and the right figure shows a log-plot with $t_0 < 0$ in Example \ref{['ex:posdim']}, and their line fittings.
  • Figure 4: Log plot of $\|U_k-U_*\|$ in Example 3.4
  • Figure 5: The left figure displays the plots of $\sqrt{k}\|U_{k}-U_*\|$ and $\sqrt[6]{k}\|U_k-U_*\|$ in Example \ref{['ex:moment']} with the initial point on the slowest curve, and the right figure displays the plots of $\|U_k - U_*\|^{-6}$ and the line fitting.

Theorems & Definitions (40)

  • Proposition 3.1
  • proof
  • Example 3.2: Known upper bounds and actual convergence rates
  • Example 3.3: Positive dimensional intersection
  • Example 3.4: Intersection with $2$-plane
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • Example 4.4
  • ...and 30 more