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.

翻译：我们研究固定阶非对称秩一尖峰张量模型的同步交替幂迭代。主要贡献是提出一种独立于特定初始化的有限迭代局部理论。一旦迭代进入种植秩一方向的足够小邻域，其误差分解为几何衰减的瞬态项和由固定在种植点的正交噪声压缩引起的固有噪声基底。确定性有限样本条件被明确表述，但在粗糙的固定阶多线性噪声事件下，这些条件简化为保守的高信号区间，适用于固定或缓慢扩展的局部半径。随后，我们将暖启动机制与任何特定谱构造分离。一个通用的单次扫描原理表明：若符号兼容的初始化器具有相关系数\(γ_N\)、首次扫描噪声水平\(a_N\)且满足\(a_N/(γ_N^{d-1}ω_{N,d})\to0\)，则可选择扩展半径\(r_N=o(ω_{N,d})\)，使得首次扫描进入局部盆地。进入后，局部仿射收缩收敛于该盆地中唯一的信息局部不动点。对于中心化Gram矩阵初始化，我们通过信号保持的消去法比较（固定尖峰并平均删除坐标）和平均消去切片收缩估计（称为后压估计），在独立同分布有限四阶矩噪声下验证所需的相关系数和同一样本首次扫描噪声界。消去法比较保持尖峰固定，对删除坐标取平均，因此种植坐标通过\(\ell_2\)加权和而非最坏情况不相关性界限进入分析。