投影幂迭代在排列同步中的恢复理论

中文摘要 AI 辅助

我们研究了投影幂方法（PPM）用于在可能稀疏的均匀损坏模型下同步n个未知的m个对象的排列。每对对象以概率p被观测，且观测测量以概率π0未被损坏，否则为独立的均匀排列。在log m=o(npπ0^2)条件下，我们证明了对于具有固定正多数正确块的独立估计，每个指定块的高概率精确一步恢复。当np≥C0 log n且m=o(npπ0^2)时，我们证明一个高概率事件同时为每个最优对齐误差至多0.5-ε的估计产生块误差收缩。收缩因子为O(m/(npπ0^2))，误差下限为O(e^{-cnpπ0}+e^{-cnpπ0^2}+log n/n)。因此，一次更新将这一盆地中每个可能依赖数据的估计映射到消失的块误差，并且所有后续迭代在迭代索引上均匀地保持几乎精确。一步和轨迹结果扩展到独立的、非同分布的、排列值损坏，其均值为m^{-1} 11^{⊤}。在均匀模型下，参考块谱初始化器具有对齐块误差O_P(m/(npπ0^2))，产生端到端的几乎精确恢复保证。在更强的全块信号条件下，PPM在有限次迭代后达到精确恢复。该理论精确转移到具有共同支持的偏排列；对于变化支持，我们建立了确定性和概率性的共视边际。

英文摘要

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.

发表机构

• Saint Louis University（圣路易斯大学）
• University of Minnesota（明尼苏达大学）

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