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Tensor CANDECOMP/PARAFAC (CP) decomposition is an important tool that solvesa wide class of machine learning problems. Existing popular approaches recovercomponents one by one, not necessarily in the order of larger components first.Recently developed simultaneous power method obtains only a high probabilityrecovery of top $r$ components even when the observed tensor is noiseless. Wepropose a Slicing Initialized Alternating Subspace Iteration (s-ASI) methodthat 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 methodintroduces a Slice-Based Initialization that runs$O(1/\log(\frac{\lambda_r}{\lambda_{r+1}}))$ steps of matrix subspaceiterations, where $\lambda_r$ denotes the r-th top singular value of thetensor. We are the first to provide a theoretical guarantee on simultaneousorthogonal asymmetric tensor decomposition. Under the noiseless case, we arethe first to provide an \emph{almost sure} theoretical guarantee onsimultaneous orthogonal tensor decomposition. When tensor is noisy, ouralgorithm 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 asmall constant proportional to the probability of bad initializations in thenoisy setting.

Guaranteed Simultaneous Asymmetric Tensor Decomposition via Orthogonalized Alternating Least Squares