Generalization of Hinging Hyperplanes

The model of hinging hyperplanes (HH) can approximate a large class of nonlinear functions to arbitrary precision, but represent only a small part of continuous piecewise-linear (CPWL) functions in two or more dimensions. In this correspondence, the influence of this drawback for black-box modeling is first illustrated by a simple example. Then it is shown that the above shortcoming can be amended by adding a sufficient number of linear functions to current hinges. It is proven that any CPWL function of n variables can be represented by a sum of hinges containing at most n+1 linear functions. Hence the model of a sum of such expanded hinges is a general representation for all CPWL functions. The structure of the novel general representation is much simpler than the existing generalized canonical representation that consists of nested absolute-value functions. This characteristic is very useful for black-box modeling. Based on the new general representation, an upper bound on the number of nestings of nested absolute-value functions of a generalized canonical representation is established, which is much smaller than the known result.