We consider the Principal Component Analysis problem for large tensors of arbitrary order @math under a single-spike (or rank-one plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessary and sufficient conditions under which the principal component can be estimated using unbounded computational resources. It turns out that this is possible as soon as the signal-to-noise ratio @math becomes larger than @math (and in particular @math can remain bounded as the problem dimensions increase). On the other hand, we analyze several polynomial-time estimation algorithms, based on tensor unfolding, power iteration and message passing ideas from graphical models. We show that, unless the signal-to-noise ratio diverges in the system dimensions, none of these approaches succeeds. This is possibly related to a fundamental limitation of computationally tractable estimators for this problem. We discuss various initializations for tensor power iteration, and show that a tractable initialization based on the spectrum of the matricized tensor outperforms significantly baseline methods, statistically and computationally. Finally, we consider the case in which additional side information is available about the unknown signal. We characterize the amount of side information that allows the iterative algorithms to converge to a good estimate.
As mentioned above, power iteration is a natural approach to tensor factorization and was studied in several earlier papers. Most recently, interest within machine learning was spurred by @cite_22 .
@cite_22
2018776447
In this paper we formulate multi-target tracking (MTT) as a rank-1 tensor approximation problem and propose an l1 norm tensor power iteration solution. In particular, a high order tensor is constructed based on trajectories in the time window, with each tensor element as the affinity of the corresponding trajectory candidate. The local assignment variables are the l1 normalized vectors, which are used to approximate the rank-1 tensor. Our approach provides a flexible and effective formulation where both pairwise and high-order association energies can be used expediently. We also show the close relation between our formulation and the multi-dimensional assignment (MDA) model. To solve the optimization in the rank-1 tensor approximation, we propose an algorithm that iteratively powers the intermediate solution followed by an l1 normalization. Aside from effectively capturing high-order motion information, the proposed solver runs efficiently with proved convergence. The experimental validations are conducted on two challenging datasets and our method demonstrates promising performances on both.