Fast algorithms for Higher-order Singular Value Decomposition from incomplete data

TL;DR

This paper introduces efficient algorithms for computing the higher-order singular value decomposition (HOSVD) from incomplete data by solving a unified optimization problem, improving accuracy and convergence over existing methods.

Contribution

The paper formulates a single optimization framework for incomplete HOSVD, introduces two algorithms with proven global convergence, and demonstrates superior performance in practical applications.

Findings

Algorithms outperform state-of-the-art methods in accuracy

Global convergence of proposed algorithms is established

Practical applications show improved results in face recognition and MRI reconstruction

Abstract

Higher-order singular value decomposition (HOSVD) is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present two algorithms for solving the problem based on block coordinate update. Global convergence of both algorithms is shown under mild assumptions. The convergence of the second algorithm implies that of the popular higher-order orthogonality iteration (HOOI) method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed methods to state-of-the-art ones for solving incomplete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our methods over other compared ones. Furthermore, we apply them to face recognition and MRI image reconstruction to show their practical performance.