Logistic Localized Modeling of the Sample Space for Feature Selection and Classification

Conventional feature selection algorithms assign a single common feature set to all regions of the sample space. In contrast, this paper proposes a novel algorithm for localized feature selection for which each region of the sample space is characterized by its individual distinct feature subset that may vary in size and membership. This approach can therefore select an optimal feature subset that adapts to local variations of the sample space, and hence offer the potential for improved performance. Feature subsets are computed by choosing an optimal coordinate space so that, within a localized region, within-class distances and between-class distances are, respectively, minimized and maximized. Distances are measured using a logistic function metric within the corresponding region. This enables the optimization process to focus on a localized region within the sample space. A local classification approach is utilized for measuring the similarity of a new input data point to each class. The proposed logistic localized feature selection (lLFS) algorithm is invariant to the underlying probability distribution of the data; hence, it is appropriate when the data are distributed on a nonlinear or disjoint manifold. lLFS is efficiently formulated as a joint convex/increasing quasi-convex optimization problem with a unique global optimum point. The method is most applicable when the number of available training samples is small. The performance of the proposed localized method is successfully demonstrated on a large variety of data sets. We demonstrate that the number of features selected by the lLFS method saturates at the number of available discriminative features. In addition, we have shown that the Vapnik-Chervonenkis dimension of the localized classifier is finite. Both these factors suggest that the lLFS method is insensitive to the overfitting issue, relative to other methods.