{"ID":3052969,"CreatedAt":"2026-06-04T04:41:36.695875263Z","UpdatedAt":"2026-06-05T11:43:53.432517148Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04065","arxiv_id":"2606.04065","title":"Finite-Iteration Local Dynamics and Warm Starts for Alternating Power Iteration in Spiked Tensor PCA","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.","short_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 er...","url_abs":"https://arxiv.org/abs/2606.04065","url_pdf":"https://arxiv.org/pdf/2606.04065v1","authors":"[\"Yanjin Xiang\",\"Zhihua Zhang\"]","published":"2026-06-02T13:44:12Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.ST\"]","methods":"[]","has_code":false}