level: research
this paper studies alternating power iteration for rank-one spiked tensor models with fixed order and asymmetric structure. the main result is a finite-iteration local theory that does not depend on any particular starting point. once the iterates land in a small enough neighborhood of the true signal direction, the error splits into two parts: a transient that shrinks geometrically fast and a steady noise floor from fixed orthogonal noise contractions at the planted point.
the analysis gives explicit finite-sample conditions. under a coarse fixed-order multilinear noise event, these conditions simplify to a high-signal regime for fixed or slowly growing local radii. the work then separates the warm-start mechanism from any specific spectral initialization method. a generic one-sweep principle shows that if a sign-compatible initializer has correlation gamma_n and first-sweep noise level a_n, and a_n divided by gamma_n to the power d-1 times omega_{n,d} goes to zero, then the method succeeds.
the results provide a clearer understanding of how alternating power iteration behaves in tensor pca. by decoupling the local convergence from the initialization, the theory offers practical guidance on when warm starts are effective. the finite-iteration bounds are useful for algorithm design and for predicting performance in high-dimensional tensor problems common in machine learning and statistics.
why it matters: this theory helps practitioners choose and analyze initialization methods for tensor decomposition, a key task in unsupervised learning and multiway data analysis.