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Revisiting Trace Norm Minimization for Tensor Tucker Completion: A Direct Multilinear Rank Learning Approach

Xueke Tong, Hancheng Zhu, Lei Cheng, Yik-Chung Wu

TL;DR

This work tackles Tucker tensor completion by addressing the challenge of learning the multilinear rank. It introduces a CPD-based equivalent Tucker representation in which the core's low-rank structure is enforced directly on the factor matrices $\boldsymbol B^{(k)}=\boldsymbol A^{(k)}\boldsymbol \Xi^{(k)}$, forming the LRFMTC framework. An auxiliary-variable-free optimization problem is derived and solved via a convergent block coordinate descent with accelerated fixed-point iterations, providing convergence guarantees to a KKT point. Empirically, LRFMTC yields more accurate multilinear rank estimation and smaller tensor completion errors than state-of-the-art Tucker completion methods across synthetic, image, and chemometrics data, with competitive runtimes. This approach offers a direct, scalable path to multilinear rank learning and suggests avenues for incorporating other low-rank regularizers.

Abstract

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.

Revisiting Trace Norm Minimization for Tensor Tucker Completion: A Direct Multilinear Rank Learning Approach

TL;DR

This work tackles Tucker tensor completion by addressing the challenge of learning the multilinear rank. It introduces a CPD-based equivalent Tucker representation in which the core's low-rank structure is enforced directly on the factor matrices , forming the LRFMTC framework. An auxiliary-variable-free optimization problem is derived and solved via a convergent block coordinate descent with accelerated fixed-point iterations, providing convergence guarantees to a KKT point. Empirically, LRFMTC yields more accurate multilinear rank estimation and smaller tensor completion errors than state-of-the-art Tucker completion methods across synthetic, image, and chemometrics data, with competitive runtimes. This approach offers a direct, scalable path to multilinear rank learning and suggests avenues for incorporating other low-rank regularizers.

Abstract

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.
Paper Structure (9 sections, 33 equations, 8 figures, 9 tables, 1 algorithm)

This paper contains 9 sections, 33 equations, 8 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Tucker Completion and Rationale of Previous Optimization Modeling
  3. Low-rank Factor Matrices Based Tucker Trace Norm Minimization
  4. Auxiliary Variables-free Optimization Algorithm
  5. Experimental Results
  6. Synthetic data
  7. Image data
  8. Chemometrics data
  9. Conclusion

Figures (8)

  • Figure 1: Averaged RSE comparison under a) sampling ratio (SR)=20% at SNR=20dB and b) sampling ratio (SR)=10% at no noise. The vertical error bars show one standard deviation.
  • Figure 2: Averaged RSE comparison on a) Amino Acids Fluorescence and b) Sugar Process at SNR=20dB. The vertical error bars show one standard deviation of the RSE across multiple trials.
  • Figure 3: RSE comparison on synthetic data with SR=20% and SNR=20dB
  • Figure 4: RSE comparison on color imate with SR=30% and SNR=20dB
  • Figure 5: RSE of synthetic data in SR=20% and SNR=20dB
  • ...and 3 more figures