Symmetric tensor decomposition by alternating gradient descent

• Published in: Numerical linear algebra with applications Vol. 29; no. 1
• Main Author: Liu, Haixia
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.01.2022
• Subjects: Algorithms; alternating minimization; Convergence; CP rank; Decomposition; gradient descent; Greedy algorithms; linear convergence; Mathematical analysis; Optimization; symmetric tensor decomposition; Tensors
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2406

The symmetric tensor decomposition problem is a fundamental problem in many fields, which appealing for investigation. In general, greedy algorithm is used for tensor decomposition. That is, we first find the largest singular value and singular vector and subtract the corresponding component from tensor, then repeat the process. In this article, we focus on designing one effective algorithm and giving its convergence analysis. We introduce an exceedingly simple and fast algorithm for rank‐one approximation of symmetric tensor decomposition. Throughout variable splitting, we solve symmetric tensor decomposition problem by minimizing a multiconvex optimization problem. We use alternating gradient descent algorithm to solve. Although we focus on symmetric tensors in this article, the method can be extended to nonsymmetric tensors in some cases. Additionally, we also give some theoretical analysis about our alternating gradient descent algorithm. We prove that alternating gradient descent algorithm converges linearly to global minimizer. We also provide numerical results to show the effectiveness of the algorithm.