Feature Transformation by Function Decomposition

While not explicitly intended for feature transformation, some methods for switching circuit design implicitly deal with this problem. Given a tabulated Boolean function, these methods construct a circuit that implements that function. In 1950s and 1960s, Ashenhurst [1] and Curtis [2] proposed a function decomposition method that develops a switching circuit by constructing a nested hierarchy of tabulated Boolean functions. Both the hierarchy and the functions themselves are discovered by the decomposition method and are not given in advance. This is especially important from the viewpoint of feature construction, since the outputs of such functions can be regarded as new features not present in the original problem description. The basic principle of function decomposition is the following. Let a tabulated function y = F (X) use a set of input features X = x1, . . . , xn. The goal is to decompose this function into y = G(A, H(B)), where A and B are subsets of features in X such that A ∪ B = X. G and H are tabulated functions that are determined by the decomposition and are not predefined. Their joint complexity (determined by some complexity measure) should be lower than the complexity of F . Such a decomposition also discovers a new feature c = H(B). Since the decomposition can be applied recursively on H and G, the result in general is a hierarchy of features. For each feature in the hierarchy, there is a corresponding tabulated function (such as H(B)) that determines the dependency of that feature on its immediate descendants in the hierarchy. Ashenhurst-Curtis decomposition was intended for switching circuit design of completely specified Boolean functions. Recently, Wan and Perkowski [3] extended the decomposition to handle incompletely specified Boolean functions. In the framework of feature extraction, Ross et al. [4] used a set of simple Boolean functions to show the decomposition’s capability to discover and extract useful features. This article presents an approach to feature transfor-