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Data-Adaptive Transformed Bilateral Tensor Low-Rank Representation for Clustering

Hui Chen, Xinjie Wang, Xianchao Xiu, Wanquan Liu

TL;DR

The paper addresses robustness gaps in tensor low-rank representation for clustering by introducing TBTLRR, a data-adaptive transformed bilateral tensor LRR framework. It learns a unitary transform along the third mode, uses the transformed tensor nuclear norm, and combines $\ell_{1/2}$ sparsity with Frobenius noise regularization to handle mixed noise. An ADMM-based optimization with subspace projections solves the nonconvex model, and a diagonal-ratio slice weighting strategy improves affinity construction. Extensive experiments on multiple benchmarks show superior clustering accuracy and stability, along with favorable computational efficiency. The work advances robust subspace clustering by integrating learnable transforms, bilateral tensor modeling, and targeted regularization into a unified, scalable framework.

Abstract

Tensor low-rank representation (TLRR) has demonstrated significant success in image clustering. However, most existing methods rely on fixed transformations and suffer from poor robustness to noise. In this paper, we propose a novel transformed bilateral tensor low-rank representation model called TBTLRR, which introduces a data-adaptive tensor nuclear norm by learning arbitrary unitary transforms, allowing for more effective capture of global correlations. In addition, by leveraging the bilateral structure of latent tensor data, TBTLRR is able to exploit local correlations between image samples and features. Furthermore, TBTLRR integrates the $\ell_{1/2}$-norm and Frobenius norm regularization terms for better dealing with complex noise in real-world scenarios. To solve the proposed nonconvex model, we develop an efficient optimization algorithm inspired by the alternating direction method of multipliers (ADMM) and provide theoretical convergence. Extensive experiments validate its superiority over the state-of-the-art methods in clustering. The code will be available at https://github.com/xianchaoxiu/TBTLRR.

Data-Adaptive Transformed Bilateral Tensor Low-Rank Representation for Clustering

TL;DR

The paper addresses robustness gaps in tensor low-rank representation for clustering by introducing TBTLRR, a data-adaptive transformed bilateral tensor LRR framework. It learns a unitary transform along the third mode, uses the transformed tensor nuclear norm, and combines sparsity with Frobenius noise regularization to handle mixed noise. An ADMM-based optimization with subspace projections solves the nonconvex model, and a diagonal-ratio slice weighting strategy improves affinity construction. Extensive experiments on multiple benchmarks show superior clustering accuracy and stability, along with favorable computational efficiency. The work advances robust subspace clustering by integrating learnable transforms, bilateral tensor modeling, and targeted regularization into a unified, scalable framework.

Abstract

Tensor low-rank representation (TLRR) has demonstrated significant success in image clustering. However, most existing methods rely on fixed transformations and suffer from poor robustness to noise. In this paper, we propose a novel transformed bilateral tensor low-rank representation model called TBTLRR, which introduces a data-adaptive tensor nuclear norm by learning arbitrary unitary transforms, allowing for more effective capture of global correlations. In addition, by leveraging the bilateral structure of latent tensor data, TBTLRR is able to exploit local correlations between image samples and features. Furthermore, TBTLRR integrates the -norm and Frobenius norm regularization terms for better dealing with complex noise in real-world scenarios. To solve the proposed nonconvex model, we develop an efficient optimization algorithm inspired by the alternating direction method of multipliers (ADMM) and provide theoretical convergence. Extensive experiments validate its superiority over the state-of-the-art methods in clustering. The code will be available at https://github.com/xianchaoxiu/TBTLRR.
Paper Structure (29 sections, 2 theorems, 49 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 49 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Definitions
  4. Related Work
  5. Methodology
  6. New Model
  7. Optimization Algorithm
  8. Update $\mathcal{J}$ while fix others
  9. Update $\bar{\mathcal{Z}}$ while fix others
  10. Update $\mathcal{T}$ while fix others
  11. Update $\bar{\mathcal{L}}$ while fix others
  12. Update $\mathcal{N}$ while fix others
  13. Update $\mathcal{E}$ while fix others
  14. Update multipliers $\mathcal{P}_{t+1}, \mathcal{G}_{t+1}, \mathcal{W}_{t+1}$ by
  15. Convergence Analysis
  16. ...and 14 more sections

Key Result

Lemma 3.2

Assume that $\{\mathcal{H}_t\}$ is a sequence generated by Algorithm algorithm：1. Then $\{\mathcal{H}_{t}\}$ is bounded.

Figures (10)

  • Figure 1: Overview of the proposed TBTLRR. Firstly, TRPCA is used to remove noise from the input data. Secondly, two dictionary tensors are constructed via the T-TSVD. Thirdly, the proposed TBTLRR decomposes the data into low-rank, sparse, and Gaussian noise tensors. Finally, clustering is performed on the coefficient tensor to obtain the final results.
  • Figure 2: Illustration of the low tubal-rank property, where (a)-(b) are the selected images from the Berkeley Segmentation dataset, (c)-(d) are the corresponding singular value sum distributions.
  • Figure 3: Visualization on the Umist dataset, where (a)-(b) are the representative frontal slices of the coefficient tensor, (c) is the affinity matrix from simple averaging over all slices, and (d) is the one via our proposed diagonal-ratio-based weighted fusion.
  • Figure 4: ACC (%) and NMI (%) of all compared methods on different datasets with varying proportions of sparse noise.
  • Figure 5: ACC (%) and NMI (%) of all compared methods on different datasets with varying proportions of Gaussian noise.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Definition 2.1: Unitary transform song2020robust
  • Definition 2.2: Block diagonal and fold song2020robust
  • Definition 2.3: $T$-product kernfeld2015tensor
  • Definition 2.4: Orthogonal tensor kernfeld2015tensor
  • Definition 2.5: $f$-diagonal kolda2009tensor
  • Definition 2.6: TNN kolda2009tensor
  • Definition 2.7: T-TSVD kernfeld2015tensor
  • Definition 2.8: TTNN kernfeld2015tensor
  • Remark 3.1
  • Lemma 3.2
  • ...and 1 more