Exactly Robust Kernel Principal Component Analysis.

Area of Science:

• Machine Learning
• Data Science
• Matrix Decomposition

Background:

• Robust Principal Component Analysis (RPCA) excels at recovering low-rank matrices corrupted by sparse noise.
• However, RPCA is ineffective for high-rank matrices, a common scenario in real-world data.
• Existing methods lack robustness to sparse noise for nonlinear, high-rank matrix decomposition.

Purpose of the Study:

• To introduce Robust Kernel Principal Component Analysis (RKPCA), a novel unsupervised nonlinear method.
• To enable the decomposition of partially corrupted matrices into sparse and low-dimensional latent components.
• To address the limitations of RPCA in handling high-rank matrices with sparse noise.

Main Methods:

• RKPCA decomposes matrices into sparse and low-rank components, accommodating high or full-rank matrices.
• The method employs two novel nonconvex optimization algorithms: alternating direction method of multipliers with backtracking line search and proximal linearized minimization with adaptive step size (AdSS).
• Theoretical analysis guarantees high recovery accuracy with high probability.

Main Results:

• RKPCA demonstrates superior performance in noise removal compared to existing methods.
• Robust subspace clustering using RKPCA shows significant effectiveness and superiority.
• The method achieves high recovery accuracy on corrupted, high-rank matrices.

Conclusions:

• RKPCA is the sole unsupervised nonlinear method robust to sparse noise for matrix decomposition.
• The proposed optimization algorithms effectively handle the nonconvex and indifferentiable problems inherent in RKPCA.
• RKPCA offers a powerful and versatile solution for noise removal and robust subspace clustering with high-rank matrices.