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32] where KPCA outperforms PCA in low-dimensional manifolds). However, Bayesian matching achieves ≈90% with only four projections— two for each P (∆ | Ω)—and dominates both PCA and KPCA throughout the entire range of subspace dimensions in Fig. 12. 2. 2 Comparison of the subspace techniques across multiple attributes (k = 20) G. Shakhnarovich and B. Moghaddam PCA ICA KPCA Bayes Accuracy 77% 77% 87% 95% Computation 108 109 109 108 Uniqueness Yes No Yes Yes Projections Linear Linear Nonlinear Linear repeatable), whereas ICA is not unique owing to the variety of techniques used to compute the basis and the iterative (stochastic) optimizations involved.
Subsequently, the KPCA principal components of any input vector can be efficiently computed with simple kernel evaluations against the dataset. 25) i=1 where V n is the nth eigenvector of the feature space defined by Ψ . As with PCA, the eigenvectors V n can be ranked by decreasing order of their eigenvalues λn and a d-dimensional manifold projection of x is y = (y1 , . . 25). A significant advantage of KPCA over neural network and principal curves is that KPCA does not require nonlinear optimization, is not subject to overfitting, and does not require prior knowledge of network architecture or the number of dimensions.
These properties are illustrated in Fig. 1, where PCA is successful in finding the principal manifold, and in Fig. 8a (see later), where it is less successful, owing to clear nonlinearity of the principal manifold. PCA may be implemented via singular value decomposition (SVD). 4) 2 Face Recognition in Subspaces 23 where the M × N matrix U and the N × N matrix V have orthonormal columns, and the N × N matrix D has the singular values1 of X on its main diagonal and zero elsewhere. 2). When the number of examples M is much smaller than the dimension N , this is a crucial advantage.