Revisiting trace norm minimization for tensor Tucker completion: A direct multilinear rank learning approach

• Published in: Pattern recognition Vol. 158; p. 110995
• Main Authors: Tong, Xueke, Zhu, Hancheng, Cheng, Lei, Wu, Yik-Chung
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.02.2025
• Subjects: Multilinear rank; Tensor decomposition; Trace norm minimization; Tucker model; Tensor decomposition; Tucker model; Multilinear rank; Trace norm minimization
• ISSN: 0031-3203
• DOI: 10.1016/j.patcog.2024.110995

To efficiently express tensor data using the Tucker format, a critical task is to minimize the multilinear rank such that the model would not be over-flexible and lead to overfitting. Due to the lack of rank minimization tools in tensor, existing works connect Tucker multilinear rank minimization to trace norm minimization of matrices unfolded from the tensor data. While these formulations try to exploit the common aim of identifying the low-dimensional structure of the tensor and matrix, this paper reveals that existing trace norm-based formulations in Tucker completion are inefficient in multilinear rank minimization. We further propose a new interpretation of Tucker format such that trace norm minimization is applied to the factor matrices of the equivalent representation, rather than some matrices unfolded from tensor data. Based on the newly established problem formulation, a fixed point iteration algorithm is proposed, and its convergence is proved. Numerical results are presented to show that the proposed algorithm exhibits significant improved performance in terms of multilinear rank learning and consequently tensor signal recovery accuracy, compared to existing trace norm based Tucker completion methods. •An equivalent Tucker model in the form of a CPD with low-rank factor matrices.•New trace norm formulation for direct multilinear rank minimization in Tucker format.•Optimization algorithm with multilinear rank estimation.•Complexity analysis and efficiency improvement for the algorithm.•Extensive experimental results on synthetic data and real-world datasets.