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Low Rank Support Quaternion Matrix Machine

Wang Chen, Ziyan Luo, Shuangyue Wang

TL;DR

The paper introduces LSQMM, a quaternion-based classifier for color images that preserves inter-channel coupling by treating RGB channels as pure quaternions and promotes low-rank structure via a quaternion nuclear-norm regularization. An ADMM-based optimization framework solves the resulting nontrivial objective, with convergence guaranteed by a real-isomorphism transformation. Empirical results on six color-image datasets show LSQMM achieves superior accuracy and robustness, particularly in small-sample, high-dimensional settings, albeit with moderate computational cost. The work highlights the practical benefits of quaternion representations for color image classification and outlines avenues for sparsity, tensor extensions, and faster quaternion-SVD techniques.

Abstract

Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.

Low Rank Support Quaternion Matrix Machine

TL;DR

The paper introduces LSQMM, a quaternion-based classifier for color images that preserves inter-channel coupling by treating RGB channels as pure quaternions and promotes low-rank structure via a quaternion nuclear-norm regularization. An ADMM-based optimization framework solves the resulting nontrivial objective, with convergence guaranteed by a real-isomorphism transformation. Empirical results on six color-image datasets show LSQMM achieves superior accuracy and robustness, particularly in small-sample, high-dimensional settings, albeit with moderate computational cost. The work highlights the practical benefits of quaternion representations for color image classification and outlines avenues for sparsity, tensor extensions, and faster quaternion-SVD techniques.

Abstract

Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.

Paper Structure

This paper contains 14 sections, 2 theorems, 21 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Low-rank Support Quaternion Matrix Machine
  4. The LSQMM Model
  5. Optimization Algorithm
  6. Complexity Analysis
  7. Convergence Analysis
  8. Numerical Experiments
  9. Dataset Description
  10. Classification Accuracy
  11. Runtime Efficiency
  12. Noise Robustness
  13. Parameter Sensitivity
  14. Conclusion

Key Result

Theorem 2.1

Let $\mathbf{A}\in \mathbb{Q}^{m \times n}$ be of rank $r$, there exist unitary quaternion matrices $\mathbf{U} = [u_1, u_2, \dots, u_m] \in \mathbb{Q}^{m \times m}$ and $\mathbf{V} = [v_1, v_2, \dots, v_n] \in \mathbb{Q}^{n \times n}$ satisfying where $\Sigma = \operatorname{diag}(\sigma_1, \dots, \sigma_r, 0, \dots, 0) \in \mathbb{R}^{m \times n}$, and $\sigma_i$ ($i = 1, \dots, r$) represent t

Figures (3)

  • Figure 1: The image examples from the proposed dataset.
  • Figure 2: Comparison of classification accuracy across different regularization parameter ($\lambda$) .
  • Figure 3: The values of objective functions versus the number of iterations on the dataset.(a) the Eye Diseases dataset,(b) the Oxford Flowers dataset .

Theorems & Definitions (5)

  • Definition 2.1: zhang1997qsvd
  • Definition 2.2
  • Theorem 2.1: Quaternion Singular Value Decomposition (QSVD)zhang1997qsvd
  • Definition 2.3: wei2018quaternion
  • Lemma 1: Real Representation of Quaternion Matrices zhang1997qsvdwei2018quaternion