Piecewise‐linear division of region by neural networks with maximum detectors

A neural network with maximum detectors divides a space which has n‐dimensions Rⁿ into piece‐wise‐linear regions. However, there have been very limited theoretical researches concerning what kind of piecewise‐linear region divisions can be realized by a neural network, that is, concerning the formation capability of piecewise‐linear discriminate regions by a neural network. In this paper, this problem is considered theoretically. The concept of a k‐dividual region is introduced and then it is proved that a division can be realized by a piecewise‐linear machine if and only if it is k‐dividual.