Finite-Iteration Local Dynamics and Warm Starts for Alternating Power Iteration in Spiked Tensor PCA

2026-06-02

AI Summary

This work addresses the finite-step local convergence and warm-start problem of alternating power iteration in high-order asymmetric rank-one spiked tensor principal component analysis. It establishes a local dynamics theory that does not rely on specific initialization schemes. The authors propose a universal single-pass warm-start principle independent of spectral constructions, integrating techniques such as "push-back" estimation, leave-one-out coordinate averaging, ℓ²-weighted summation, and orthogonal noise shrinkage. Under only a finite fourth-moment noise assumption, this approach effectively circumvents coherence restrictions and precisely characterizes the influence of initialization and noise. Under high signal-to-noise ratios or slowly expanding local radii, the iteration is shown to enter the basin of attraction and converge to a unique informative fixed point, with the final error governed by both a geometrically decaying term and a noise floor.

Abstract

We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models. Our main contribution is a finite-iteration local theory that is independent of any particular initialization. Once the iterates enter a sufficiently small neighborhood of the planted rank-one direction, their error decomposes into a geometrically decaying transient and an intrinsic noise floor caused by fixed orthogonal noise contractions at the planted point. The deterministic finite-sample conditions are stated explicitly, but under a coarse fixed-order multilinear noise event they reduce to a conservative high-signal regime for fixed or slowly expanding local radii. We then separate the warm-start mechanism from any specific spectral construction. A generic one-sweep principle shows that, if a sign-compatible initializer has correlation \(γ_N\), first-sweep noise level \(a_N\), and \(a_N/(γ_N^{d-1}ω_{N,d})\to0\), then one can choose an expanding radius \(r_N=o(ω_{N,d})\) for which the first sweep enters the local basin. After entry, the local affine contraction yields convergence to the unique informative local fixed point in that basin. For centered-Gram initialization, we verify the required correlation and same-sample first-sweep noise bound under i.i.d. finite-fourth-moment noise by a signal-preserving noise-only leave-one comparison and an averaged leave-one slice-contraction estimate, which we call a pressed-back estimate. The leave-one comparison keeps the spike fixed and averages over the deleted coordinate, so planted coordinates enter through \(\ell_2\)-weighted sums rather than worst-case incoherence bounds.

Problem

• spiked tensor PCA
• alternating power iteration
• finite-iteration local dynamics
• warm starts
• local convergence

Innovation

• finite-iteration local dynamics
• alternating power iteration
• spiked tensor PCA
• warm start
• pressed-back estimate