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

TL;DR

This work studies convergence of alternating projections between the positive semidefinite cone and a line E = {A + tB} under the singleton intersection condition E ∩ S_+^n = {A}. By expanding the characteristic polynomial of the perturbed matrix A + tB via adjugates and compound matrices and applying the Newton diagram/polytope framework, the authors bound the leading eigenvalue terms that govern convergence. They prove that the iteration converges with rate ∥U_k − A∥ = O(k^{-1/2}) independent of singularity degree, and they provide a sufficient rank-condition for linear convergence, with a discussion of tightness in singularity-degree-2 cases. The approach bypasses conventional error-bound arguments and directly analyzes the defining projection equations through eigenvalue dynamics, offering a precise link between perturbation theory, Newton-type diagrams, and projection methods in semidefinite settings.

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.

Figures (8)

• Figure 1: $D_0$ and the Newton polytope, where $Q$ is $(2n-2m+r,2m-r-n)$. The left and the right figures correspond to the case $m-r < n-m$ and to the case $m-r> n-m$ respectively.
• Figure 2: the Newton polytope in Example \ref{['ex:newton_polytope']}
• Figure 3: the Newton polytope in a degenerate case
• Figure 4: The Newton diagram associated with the characteristic polynomial, which is obtained by rotating the left of Figure \ref{['fig:newton_polygon']}.
• Figure 5: The Newton diagram associated with the characteristic polynomial in Example \ref{['ex:diagram']}. The case of $c\neq0$ corresponds to the left figure. The point at $(2,3)$ might be located above $\Gamma_1$. The case $c=0$ with $b = 0$ corresponds to the center figure, $c=0$ with $b \neq 0$ corresponds to the right figure.

Theorems & Definitions (35)

• Theorem 1.1
• Example 2.1
• Example 2.2
• Example 2.3
• Lemma 3.1: HJ
• Proposition 3.2
• Example 3.3
• proof : Proof of Proposition $\ref{['prop:expansion']}$
• Lemma 3.4
• proof