Non-Linear Feature Extraction by Linear Principal Component Analysis Using Local Kernel

In the last decade, the effectiveness of kernel-based methods for object detection and recognition have been reported Fukui et al. (2006); Hotta (2008c); Kim et al. (2002); Pontil & Verri (1998); Shawe-Taylor & Cristianini (2004); Yang (2002). In particular, Kernel Principal Component Analysis (KPCA) took the place of traditional linear PCA as the first feature extraction step in various researches and applications. KPCA can cope with non-linear variations well. However, KPCAmust solve the eigen value problem with the number of samples × the number of samples. In addition, the computation of kernel functions with all training samples are required to map a test sample to the subspace obtained by KPCA. Therefore, the computational cost is the main drawback. To reduce the computational cost of KPCA, sparse KPCA Tipping (2001) and the use of clustering Ichino et al. (2007 (in Japanese) were proposed. Ichino et al. Ichino et al. (2007 (in Japanese) reported that KPCA of cluster centers is more effective than sparse KPCA. However, the computational cost becomes a big problem again when the number of classes is large and each class has one subspace. For example, KPCA of visual words (cluster centers of local features) Hotta (2008b) was effective for object categorization but the computational cost is high. In this method, each category of 101 categories has one subspace constructed by 400 visual words. Namely, 40, 400 (= 101 categorizes × 400 visual words) kernel computations are required to map a local feature to all subspaces. On the other hand, traditional linear PCA is independent of the number of samples when the dimension of a feature is smaller than the number of samples. This is because the size of eigen value problem depends on the minimum number of the feature dimension and the number of samples. To map a test sample to a subspace, only inner products between basis vectors and the test sample are required. Therefore, in general, the computational cost of linear PCA is much lower than KPCA. In this paper, we propose how to use non-linearity of KPCA and computational cost of linear PCA simultaneously Hotta (2008a). Kernel-based methods map training samples to high dimensional space as x → φ(x). Nonlinearity is realized by linear method in high dimensional space. The dimension of mapped feature space of the Radial Basis Function (RBF) kernel becomes infinity, and we can not describe the mapped feature explicitly. However, the mapped feature φ(x) of the polynomial kernel can be described explicitly. This means that KPCA with the polynomial kernel can be solved directly by linear PCA of mapped features. Unfortunately, in general, the dimension of mapped features is too high to solve by linear PCA even if the polynomial kernel with 2nd degrees K(x, y) = (1+ xTy)2 is used. The dimension of mapped features of the polynomial 5