A Geometric Theory of Feature Selection and Distance-Based Measures

Abstract

Feature selection measures are often explained by the analogy to a rule to measure the “distance” of sets of features to the “closest” ideal sets of features. An ideal feature set is such that it can determine classes uniquely and correctly. This way of explanation was just an analogy before this paper. In this paper, we show a way to map arbitrary feature sets of datasets into a common metric space, which is indexed by a real number p with 1 ≤ p ≤ ∞. Since this determines the distance between an arbitrary pair of feature sets, even if they belong to different datasets, the distance of a feature set to the closest ideal feature set can be used as a feature selection measure. Surprisingly, when p = 1, the measure is identical to the Bayesian risk, which is probably the feature selection measure that is used the most widely in the literature. For 1 < p ≤ ∞, the measure is novel and has significantly different properties from the Bayesian risk. We also investigate the correlation between measurements by these measures and classification accuracy through experiments. As a result, we show that our novel measures with p > 1 exhibit stronger correlation than the Bayesian risk.