PCA and kernel PCA using polynomial filtering: a case study on face recognition∗

Principal component analysis (PCA) is an extensively used dimensionality reduction technique, with important applications in many fields such as pattern recognition, computer vision and statistics. It employs the eigenvectors of the covariance matrix of the data to project it on a lower dimensional subspace. Kernel PCA, a generalized version of PCA, performs PCA implicitly in a nonlinearly transformed feature space. In many cases, experiments show that kernel PCA is more effective than conventional PCA. However, the requirement of PCA eigenvectors is a computational bottleneck which poses serious challenges and limits the applicability of PCA-based methods, especially for real-time computations. This paper proposes an alternative framework, relying on polynomial filtering which enables efficient implementations of both PCA and its kernelized version. Further improvements are achieved when polynomial filtering is combined with wavelet transforms to obtain sparse representations of images. We showcase the applicability of the proposed scheme on face recognition. In particular, we consider the eigenfaces and kernel eigenfaces methods which employ PCA and kernel PCA respectively. The numerical experiments reported indicate that the proposed technique competes with the PCA based methods in terms of recognition rate, while being much more efficient in terms of computational and storage cost.