提出一种无需特定初始化的张量主成分分析迭代方法，可快速收敛至真实信号方向。
Finite-Iteration Local Dynamics and Warm Starts for Alternating Power Iteration in Spiked Tensor PCA
- 建立有限步数局部收敛理论，独立于初始值。
- 进入邻域后误差呈几何衰减，噪声水平由固定正交噪声决定。
- 适用于高信噪比场景，适合做算法优化与理论分析的研究者。
我们研究固定阶数非对称秩一扰动张量模型下的联合交替幂迭代。主要贡献是建立一个不依赖特定初始化的有限步数局部理论：一旦迭代点进入足够小的植根方向邻域，其误差分解为几何衰减的瞬态项和由植根点处固定正交噪声收缩引起的内在噪声底限。确定性有限样本条件明确给出，但在粗略的固定阶多重线性噪声事件下，简化为固定或缓慢扩展局部半径下的保守高信噪比情形。随后将暖启动机制与特定谱构造分离。通用单轮原则表明：若符号相容初始化相关性为 $γ_N$，首轮噪声水平为 $a_N$，且 $a_N/(γ_N^{d-1}ω_{N,d})\to0$，则可选取 $r_N=o(ω_{N,d})$ 的扩展半径，使首轮进入局部吸引盆。进入后，局部仿射收缩保证收敛至该盆内唯一有信息的局部不动点。对于中心化格拉姆初始化，我们通过保持信号不变的仅噪声留一比较及平均留一切片收缩估计（称为压回估计），在独立同分布、四阶矩有限噪声下验证了所需相关性和同样本首轮噪声界。留一比较固定信号并平均删除坐标，使得植根坐标通过 $\ell_2$ 加权和进入，而非最坏情况的非相干界。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。