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Orthogonal Subspace Clustering: Enhancing High-Dimensional Data Analysis through Adaptive Dimensionality Reduction and Efficient Clustering

Qing-Yuan Wen, Da-Qing Zhang

Abstract

This paper presents Orthogonal Subspace Clustering (OSC), an innovative method for high-dimensional data clustering. We first establish a theoretical theorem proving that high-dimensional data can be decomposed into orthogonal subspaces in a statistical sense, whose form exactly matches the paradigm of Q-type factor analysis. This theorem lays a solid mathematical foundation for dimensionality reduction via matrix decomposition and factor analysis. Based on this theorem, we propose the OSC framework to address the "curse of dimensionality" -- a critical challenge that degrades clustering effectiveness due to sample sparsity and ineffective distance metrics. OSC integrates orthogonal subspace construction with classical clustering techniques, introducing a data-driven mechanism to select the subspace dimension based on cumulative variance contribution. This avoids manual selection biases while maximizing the retention of discriminative information. By projecting high-dimensional data into an uncorrelated, low-dimensional orthogonal subspace, OSC significantly improves clustering efficiency, robustness, and accuracy. Extensive experiments on various benchmark datasets demonstrate the effectiveness of OSC, with thorough analysis of evaluation metrics including Cluster Accuracy (ACC), Normalized Mutual Information (NMI), and Adjusted Rand Index (ARI) highlighting its advantages over existing methods.

Orthogonal Subspace Clustering: Enhancing High-Dimensional Data Analysis through Adaptive Dimensionality Reduction and Efficient Clustering

Abstract

This paper presents Orthogonal Subspace Clustering (OSC), an innovative method for high-dimensional data clustering. We first establish a theoretical theorem proving that high-dimensional data can be decomposed into orthogonal subspaces in a statistical sense, whose form exactly matches the paradigm of Q-type factor analysis. This theorem lays a solid mathematical foundation for dimensionality reduction via matrix decomposition and factor analysis. Based on this theorem, we propose the OSC framework to address the "curse of dimensionality" -- a critical challenge that degrades clustering effectiveness due to sample sparsity and ineffective distance metrics. OSC integrates orthogonal subspace construction with classical clustering techniques, introducing a data-driven mechanism to select the subspace dimension based on cumulative variance contribution. This avoids manual selection biases while maximizing the retention of discriminative information. By projecting high-dimensional data into an uncorrelated, low-dimensional orthogonal subspace, OSC significantly improves clustering efficiency, robustness, and accuracy. Extensive experiments on various benchmark datasets demonstrate the effectiveness of OSC, with thorough analysis of evaluation metrics including Cluster Accuracy (ACC), Normalized Mutual Information (NMI), and Adjusted Rand Index (ARI) highlighting its advantages over existing methods.
Paper Structure (14 sections, 1 theorem, 24 equations, 4 figures, 6 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 24 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Experiments and Results Analysis
  5. Experimental Setup
  6. Dataset
  7. Cluster evaluation metrics
  8. Experimental Results
  9. Experimental Analysis
  10. Sensitivity analysis of OSC algorithms' subspace dimensions
  11. Algorithm performance comparison across different sample sizes
  12. Analysis of OSC algorithm convergence speed
  13. Comparison examination of algorithm computational efficiency
  14. Conclusion

Key Result

Theorem 1

Let the observation data matrix ${Y} = [{y}_1, \dots, {y}_N] \in \mathbb{R}^{p \times N}$ have column vectors satisfying the following conditions: the samples are partitioned into $k$ disjoint clusters $C_1, \dots, C_k$ with $|C_i| = N_i$ and $\sum_{i=1}^k N_i = N$. For each cluster $C_i$, there exi and the residual covariance matrix satisfies where the error matrix ${\Delta}_N$ satisfies: ${\Del

Figures (4)

  • Figure 1: Representative samples from six image datasets (from left to right): ORL, Jaffe, MNIST, Coil20, Yale, and CASIA-FaceV5
  • Figure 2: ACC Comparison Across Datasets at Variance Thresholds
  • Figure 3: Convergence Speed of the OSC Algorithm across Datasets
  • Figure 4: Algorithm Runtime Comparison for High-Dimensional Datasets

Theorems & Definitions (1)

  • Theorem 1: Multi-Distribution Heteroscedastic Subspace Statistical Orthogonality