Semi-Supervised Kernel Principal Component Analysis

Semi-Supervised Kernel Principal Component Analysis (SS-KPCA) is a dimensionality reduction technique that integrates semi-supervised learning with kernel methods, particularly suitable for scenarios with limited labeled data but abundant unlabeled samples. Its core concept involves mapping data to a high-dimensional feature space using kernel functions, then optimizing the principal component extraction process by leveraging limited supervision information.

Unlike traditional KPCA, SS-KPCA utilizes unlabeled data through the following mechanisms: Kernel Space Construction: Processes nonlinear data using functions like Gaussian kernels, implicitly computing inner products in high-dimensional space. Supervision Signal Integration: Incorporates class constraints from labeled data into the objective function, such as intra-class compactness and inter-class separation. Graph Regularization: Leverages the manifold structure of unlabeled data by preserving local geometric relationships through Laplacian matrices.

MATLAB implementation typically involves these steps: Kernel Matrix Computation: Generates similarity matrices between samples based on selected kernel functions (e.g., RBF kernel) using functions like 'pdist2' for distance calculation and 'exp' for kernel transformation. Optimization Objective Design: Combines discriminant terms from labeled data with manifold regularization terms from unlabeled data, often implemented through weighted matrix operations. Eigenvalue Decomposition: Solves generalized eigenvalue problems using functions like 'eigs' or 'eig' to obtain projection directions corresponding to dominant eigenvectors.

This method demonstrates excellent performance in tasks like image classification and fault diagnosis, particularly when labeling costs are high, as it significantly enhances feature quality. Note that kernel parameter selection critically impacts performance, typically optimized through cross-validation techniques like 'crossval' in MATLAB.