Guaranteed Simultaneous Asymmetric Tensor Decomposition via Orthogonalized Alternating Least Squares

Abstract

Tensor CANDECOMP/PARAFAC (CP) decomposition is an important tool that solves a wide class of machine learning problems. Existing popular approaches recover components one by one, not necessarily in the order of larger components first. Recently developed simultaneous power method obtains only a high probability recovery of top $r$ components even when the observed tensor is noiseless. We propose a Slicing Initialized Alternating Subspace Iteration (s-ASI) method that is guaranteed to recover top $r$ components ($\epsilon$-close) simultaneously for (a)symmetric tensors almost surely under the noiseless case (with high probability for a bounded noise) using $O(\log(\log \frac{1}{\epsilon}))$ steps of tensor subspace iterations. Our s-ASI method introduces a Slice-Based Initialization that runs $O(1/\log(\frac{\lambda_r}{\lambda_{r+1}}))$ steps of matrix subspace iterations, where $\lambda_r$ denotes the r-th top singular value of the tensor. We are the first to provide a theoretical guarantee on simultaneous orthogonal asymmetric tensor decomposition. Under the noiseless case, we are the first to provide an \emph{almost sure} theoretical guarantee on simultaneous orthogonal tensor decomposition. When tensor is noisy, our algorithm for asymmetric tensor is robust to noise smaller than $\min\{O(\frac{(\lambda_r - \lambda_{r+1})\epsilon}{\sqrt{r}}), O(\delta_0\frac{\lambda_r -\lambda_{r+1}}{\sqrt{d}})\}$, where $\delta_0$ is a small constant proportional to the probability of bad initializations in the noisy setting.