Title: Local Dynamics in Finite Iterations and Warm-Start Strategies for Alternating Power Iteration in Spiked Tensor PCA

Abstract:

This paper investigates simultaneous alternating power iteration within the context of fixed-order, asymmetric, rank-one spiked tensor models. The primary contribution is a local theoretical framework valid for a finite number of iterations, which holds true regardless of the specific initialization method employed. We demonstrate that once the iterative process reaches a sufficiently small neighborhood surrounding the planted rank-one direction, the resulting error can be broken down into two components: a transient phase that decays geometrically and an intrinsic noise floor. This noise floor arises from fixed orthogonal noise contractions occurring at the planted point.

While we provide explicit deterministic finite-sample conditions, these simplify under a coarse, fixed-order multilinear noise event to a conservative high-signal regime, particularly when local radii remain fixed or expand slowly. Furthermore, we decouple the warm-start mechanism from any particular spectral construction. A generic one-sweep principle establishes that if an initializer shares the same sign as the signal, possessing a correlation $\gamma_N$ and a first-sweep noise level $a_N$, such that $a_N/(\gamma_N^{d-1}\omega_{N,d})\to0$, it is possible to select an expanding radius $r_N=o(\omega_{N,d})$. Under these conditions, the first sweep successfully enters the local basin. Once inside, the local affine contraction ensures convergence to the unique informative local fixed point within that basin.

For center-Gram initialization, we confirm that the necessary correlation and same-sample first-sweep noise bounds are met when noise follows an i.i.d. distribution with finite fourth moments. This verification relies on a signal-preserving, noise-only leave-one comparison technique and an averaged leave-one slice-contraction estimate, referred to as a "pressed-back estimate." In this leave-one comparison approach, the spike remains fixed while averaging occurs over the deleted coordinate. Consequently, planted coordinates are incorporated through $\ell_2$-weighted sums rather than relying on worst-case incoherence bounds.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC