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

arXiv: https://arxiv.org/abs/2606.04065 · HTML: https://mlsanity.com/paper/2606.04065/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 16 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 1 of 1)

- [Spectral norm of random tensors](https://mlsanity.com/paper/1407.1870/) (2014)

## Abstract (arXiv)

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 (γN), first-sweep noise level (aN), and (aN/(γNd-1ωN,d)→0), then one can choose an expanding radius (rN=o(ωN,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 (ℓ2)-weighted sums rather than worst-case incoherence bounds.