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iFCTN: Folding-Free Fully-Connected Tensor Network Decomposition for Tensor Completion

Ziyi Gan, Chunfeng Cui

TL;DR

This paper introduces iFCTN, a folding-free variant of the fully connected tensor network decomposition, to address the high computational cost of updating FCTN factors due to auxiliary subnetworks. By parameterizing each factor with Khatri–Rao product-based matrices, iFCTN enables unfolding-free updates while preserving cross-mode expressivity, enabling efficient tensor completion via a proximal alternating minimization algorithm with convergence guarantees. Theoretical analyses establish sub-network structure and global convergence, while comprehensive experiments on multispectral images and traffic data show state-of-the-art recovery performance at comparable computational cost. The work advances scalable tensor completion by combining KR-based factorization with a new intra-block TN framework that maintains expressive power without expensive tensor folding/unfolding operations.

Abstract

The fully-connected tensor network (FCTN) decomposition has recently exhibited strong modeling capabilities by connecting every pair of tensor factors, thereby capturing rich cross-mode correlations and maintaining invariance under mode transpositions. However, this advantage comes with an inherent limitation: updating the factors typically requires reconstructing auxiliary sub-networks, which entails extensive and cumbersome (un)folding. In this study, we propose intra-block FCTN (iFCTN) decomposition, a novel (un)folding-free variant of FCTN decomposition that streamlines computation. We parameterize each FCTN factor through Khatri-Rao products, which significantly reduces the complexity of reconstructing intermediate sub-networks and yields subproblems with well-structured coefficient matrices. Furthermore, we deploy the proposed iFCTN decomposition on the representative task of tensor completion and design an efficient proximal alternating minimization algorithm while retaining convergence guarantees. Extensive experiments demonstrate that iFCTN outperforms or matches state-of-the-art methods with comparable computational cost.

iFCTN: Folding-Free Fully-Connected Tensor Network Decomposition for Tensor Completion

TL;DR

This paper introduces iFCTN, a folding-free variant of the fully connected tensor network decomposition, to address the high computational cost of updating FCTN factors due to auxiliary subnetworks. By parameterizing each factor with Khatri–Rao product-based matrices, iFCTN enables unfolding-free updates while preserving cross-mode expressivity, enabling efficient tensor completion via a proximal alternating minimization algorithm with convergence guarantees. Theoretical analyses establish sub-network structure and global convergence, while comprehensive experiments on multispectral images and traffic data show state-of-the-art recovery performance at comparable computational cost. The work advances scalable tensor completion by combining KR-based factorization with a new intra-block TN framework that maintains expressive power without expensive tensor folding/unfolding operations.

Abstract

The fully-connected tensor network (FCTN) decomposition has recently exhibited strong modeling capabilities by connecting every pair of tensor factors, thereby capturing rich cross-mode correlations and maintaining invariance under mode transpositions. However, this advantage comes with an inherent limitation: updating the factors typically requires reconstructing auxiliary sub-networks, which entails extensive and cumbersome (un)folding. In this study, we propose intra-block FCTN (iFCTN) decomposition, a novel (un)folding-free variant of FCTN decomposition that streamlines computation. We parameterize each FCTN factor through Khatri-Rao products, which significantly reduces the complexity of reconstructing intermediate sub-networks and yields subproblems with well-structured coefficient matrices. Furthermore, we deploy the proposed iFCTN decomposition on the representative task of tensor completion and design an efficient proximal alternating minimization algorithm while retaining convergence guarantees. Extensive experiments demonstrate that iFCTN outperforms or matches state-of-the-art methods with comparable computational cost.

Paper Structure

This paper contains 19 sections, 7 theorems, 45 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. iFCTN Decomposition
  4. iFCTN Decomposition-Based TC Method
  5. Model
  6. Algorithm
  7. Complexity analysis
  8. Convergence analysis
  9. Experiments
  10. Multispectral images
  11. Traffic data
  12. Conclusion
  13. Proof of Proposition 1
  14. Proof of Theorem 1
  15. Details for Experiments
  16. ...and 4 more sections

Key Result

Proposition 1

Suppose that $\mathcal{X} = \operatorname{iFCTN} \left(\left\{\textbf{G}_{t}\right\}_{t=1}^{N}\right)$. Then, the cherry sub-network $\textbf{Z}_{k} \in \mathbb R^{\prod_{i \in [N] \setminus k} I_{i} \times \prod_{i \in [N] \setminus k} R_{k,i}}$ comprises two components: (i) the cherry factors $\{\ where $\mathcal{S}_{k}$ is an $(N-1)$ th-order tensor shown as follows, Moreover, the mode-$k$ unf

Figures (6)

  • Figure 1: A graphical representation of TT, TR, FCTN and the proposed iFCTN decompositions. Here, the blue dotted lines denote matrix multiplications or tenor contractions, and the KR product is visually depicted as the "cherry stem". In contrast to the TT, TR, and FCTN, which employ tensor factors, iFCTN represents each tensor factor as a "cherry chain", whose components ("cherries") are matrices.
  • Figure 1: The pseudo-color renderings of reconstructions (composed of the 29th, 19th, and 9th bands) and corresponding PSNRs for MSI datasets. The top, middle, and bottom rows display the flower, face, and toy with FM = $60\%$, respectively.
  • Figure 2: A comparative analysis of sub-network: FCTN versus the proposed iFCTN Decomposition.
  • Figure 2: The pseudo-color renderings of reconstructions (composed of the 29th, 19th, and 9th bands) and corresponding PSNRs for MSI datasets. The top, middle, and bottom rows display the flower, face, and toy with RM = $95\%$, respectively.
  • Figure 3: The pseudo-color renderings of reconstructions (composed of the 29th, 19th, and 9th bands) and corresponding PSNRs for MSI datasets. The first row is flower with RM = $90\%$, the second row is face with RM = $80\%$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 1: FCTN Decomposition FCTN
  • Definition 2: Mode-$k$ Cheery Tensor
  • Definition 3: Cherry Product
  • Remark 1
  • Definition 4: iFCTN Decomposition
  • Remark 2
  • Proposition 1: Mode-$k$ Cherry Sub-network
  • Theorem 1
  • Proposition 1: Mode-$k$ Cherry Sub-network
  • proof
  • ...and 8 more