Recovery Theory for Projected Power Iterations in Permutation Synchronization

Vahan Huroyan; Gilad Lerman

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arXiv · 2609.09502

Recovery Theory for Projected Power Iterations in Permutation Synchronization

Vahan Huroyan·Gilad Lerman

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Abstract

We study the projected power method (PPM) for synchronizing $n$ unknown permutations of $m$ objects under a possibly sparse uniform corruption model. Each pair is observed with probability $p$, and an observed measurement is uncorrupted with probability $π_0$ and is otherwise an independent uniform permutation. Under $\log m=o(npπ_0^2)$, we prove exact one-step recovery (with high probability) of each prescribed block for an independent estimate with a fixed positive majority of correct blocks. When $np\ge C_0\log n$ and $m=o(npπ_0^2)$, we prove that one high-probability event yields a block-error contraction simultaneously for every estimate whose optimally aligned error is at most $0.5-ε$. The contraction factor is $O(m/(npπ_0^2))$ and the error floor is $O(e^{-cnpπ_0}+e^{-cnpπ_0^2}+{\log n}/{n})$. Consequently, one update maps every possibly data-dependent estimate in this basin to vanishing block error, and all subsequent iterates remain almost exact uniformly over the iteration index. The one-step and trajectory results extend to independent, non-identically distributed, permutation-valued corruptions with mean $m^{-1} \mathbf{1}\mathbf{1}^{\top}$. Under the uniform model, a reference-block spectral initializer has aligned block error $O_{\mathbb P}(m/(npπ_0^2))$, yielding an end-to-end almost-exact recovery guarantee. Under a stronger all-block signal condition, PPM reaches exact recovery after finitely many iterations. The theory transfers exactly to partial permutations with common support; for varying supports, we establish deterministic and probabilistic co-visibility margins.

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Vahan Huroyan, Gilad Lerman. 2026-09-08. Recovery Theory for Projected Power Iterations in Permutation Synchronization. https://arxiv.org/abs/2609.09502

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