Tensor Power Methods: Faster and Robust for Arbitrary Order

Zhao Song, Song Yue, Zhen Zhuang

Keywords: tensor power method, arbitrary order, Canonical/Polyadic decomposition

Abstract: Tensor decomposition is a fundamental method used in various areas to deal with high-dimensional data. Among the widely recognized techniques for tensor decomposition is the Canonical/Polyadic (CP) decomposition, which breaks down a tensor into a combination of rank-1 components. In this paper, we specifically focus on CP decomposition and present a novel faster robust tensor power method (TPM) for decomposing arbitrary order tensors. Our approach overcomes the limitations of existing methods that are often restricted to lower-order ($\leq 3$) tensors or require strong assumptions about the underlying data structure. By applying the sketching method, we achieve a running time of $\widetilde{O}(n^{p-1})$ per iteration of TPM on a tensor of order $p$ and dimension $n$. Furthermore, we provide a detailed analysis applicable to any $p$-th order tensor, addressing a gap in previous works. Our proposed method offers robustness and efficiency, expanding the applicability of CP decomposition to a broader class of high-dimensional data problems.

Primary Area: optimization

Submission Number: 21763