MATLAB Implementation of Kernel PCA with Code Explanations

Kernel Principal Component Analysis (KPCA) is a nonlinear dimensionality reduction method that maps original data to a high-dimensional space using kernel functions, then performs linear PCA analysis in the mapped space. This approach is particularly suitable for datasets with complex nonlinear structures, such as face recognition tasks.

### Core Concepts of Kernel PCA Kernel Selection: Common kernel functions include Gaussian kernel (RBF) and polynomial kernel, which implicitly map data to high-dimensional spaces without explicitly computing the transformed features. In MATLAB implementation, kernel functions can be defined using anonymous functions or custom functions that compute pairwise similarities between data points. Kernel Matrix Computation: Apply the kernel function to input data to calculate the kernel matrix (Gram matrix), which reflects data similarities in the high-dimensional space. The MATLAB code typically uses vectorized operations to efficiently compute the n×n kernel matrix where n is the number of samples. Kernel Matrix Centering: To ensure PCA correctness, the kernel matrix must be centered to have zero mean in the new feature space. This involves mathematical transformations that can be implemented using matrix operations in MATLAB. Eigenvalue Decomposition: Perform eigenvalue decomposition on the centered kernel matrix and select the top k eigenvectors as nonlinear principal components. MATLAB's built-in functions like `eig()` or `svd()` can be used for this decomposition. Data Projection: Project original data onto the low-dimensional space using these principal components to achieve dimensionality reduction.

### Application in Face Recognition On ORL32 and Yale32 datasets, KPCA can effectively extract nonlinear facial features, capturing influences from factors like lighting and expression changes better than traditional PCA. During implementation, kernel parameters (such as σ value for Gaussian kernel) can be adjusted through grid search or cross-validation to optimize feature extraction performance.

### Key Implementation Considerations Data Preprocessing: Standardize input data using MATLAB's `zscore` function to prevent certain features from dominating calculations due to scale differences. Kernel Matrix Optimization: Select appropriate kernel functions and parameters, which directly impact feature quality after dimensionality reduction. MATLAB's optimization toolbox can assist in parameter tuning. Computational Efficiency: KPCA has high time complexity, especially for large datasets. Approximation methods or subsampling strategies can be implemented using MATLAB's parallel computing features. Classifier Integration: Reduced-dimensional features typically require combination with classifiers like SVM (using `fitcsvm`) or KNN (using `fitcknn`) to complete recognition tasks.

The MATLAB implementation can leverage built-in matrix operations and eigenvalue decomposition functions while utilizing kernel function libraries for efficient similarity computation. For small-scale datasets like ORL32 and Yale32, this approach achieves a good balance between computational cost and recognition performance.