Sparse Kernel PCA for Outlier Detection

TL;DR

This paper introduces a novel Sparse Kernel PCA method with elastic net regularization, demonstrating its effectiveness in outlier detection with high sparsity and providing the first theoretical analysis of SKPCA's validity.

Contribution

It formulates SKPCA as a constrained optimization problem, provides a probabilistic proof of sparse solutions, and shows improved performance over existing methods in outlier detection.

Findings

Achieves nearly matching KPCA performance with less than 4% principal components.

Outperforms recent SKPCA methods in accuracy and sparsity.

Provides the first theoretical validation of SKPCA with RBF kernel.

Abstract

In this paper, we propose a new method to perform Sparse Kernel Principal Component Analysis (SKPCA) and also mathematically analyze the validity of SKPCA. We formulate SKPCA as a constrained optimization problem with elastic net regularization (Hastie et al.) in kernel feature space and solve it. We consider outlier detection (where KPCA is employed) as an application for SKPCA, using the RBF kernel. We test it on 5 real-world datasets and show that by using just 4% (or even less) of the principal components (PCs), where each PC has on average less than 12% non-zero elements in the worst case among all 5 datasets, we are able to nearly match and in 3 datasets even outperform KPCA. We also compare the performance of our method with a recently proposed method for SKPCA by Wang et al. and show that our method performs better in terms of both accuracy and sparsity. We also provide a novel probabilistic proof to justify the existence of sparse solutions for KPCA using the RBF kernel. To the best of our knowledge, this is the first attempt at theoretically analyzing the validity of SKPCA.