Functional Decomposition - Basic Mathematical Definition

Basic Mathematical Definition

For a multivariate function, functional decomposition generally refers to a process of identifying a set of functions such that

where is some other function. Thus, we would say that the function is decomposed into functions . This process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions into a collection of constituent functions such that

where is some other function. Decompositions of this kind are interesting and important for a wide variety of reasons. In general, functional decompositions are worthwhile when there is a certain "sparseness" in the dependency structure; that is, when constituent functions are found to depend on approximately disjoint sets of variables. Thus, for example, if we can obtain a decomposition of into a hierarchical composition of functions such that, as shown in the figure at right, this would probably be considered a highly valuable decomposition.