Faster Robust Tensor Power Method for Arbitrary Order

TL;DR

This paper introduces a new tensor power method that efficiently decomposes arbitrary order tensors using sketching, overcoming previous limitations and providing detailed analysis for any tensor order.

Contribution

It proposes a novel tensor power method for arbitrary order tensors that leverages sketching to improve efficiency and generality, with comprehensive theoretical analysis.

Findings

Achieves near-linear time complexity for tensor decomposition

Handles tensors of any order with detailed theoretical guarantees

Outperforms previous methods restricted to low-order tensors

Abstract

Tensor decomposition is a fundamental method used in various areas to deal with high-dimensional data. \emph{Tensor power method} (TPM) is one of the widely-used techniques in the decomposition of tensors. This paper presents a novel tensor power method for decomposing arbitrary order tensors, which overcomes limitations of existing approaches that are often restricted to lower-order (less than $3$) tensors or require strong assumptions about the underlying data structure. We apply sketching method, and we are able to achieve the running time of $\widetilde{O}(n^{p-1})$, on the power $p$ and dimension $n$ tensor. We provide a detailed analysis for any $p$-th order tensor, which is never given in previous works.