Generating High-Order Threshold Functions with Multiple Thresholds

TL;DR

This paper introduces a method for generating high-order threshold functions using multiple thresholds, enabling optimization of neural networks and easy derivation of related functions.

Contribution

It proposes a novel generating method for threshold functions utilizing a boundary-determining vector, facilitating the extension and optimization of high-order neural networks.

Findings

Constant functions can be generated from multithreshold threshold functions.

Functions with the same weights but different thresholds can be derived easily.

The overall network order can be extended without altering the structure.

Abstract

In this paper, we consider situations in which a given logical function is realized by a multithreshold threshold function. In such situations, constant functions can be easily obtained from multithreshold threshold functions, and therefore, we can show that it becomes possible to optimize a class of high-order neural networks. We begin by proposing a generating method for threshold functions in which we use a vector that determines the boundary between the linearly separable function and the high-order threshold function. By applying this method to high-order threshold functions, we show that functions with the same weight as, but a different threshold than, a threshold function generated by the generation process can be easily obtained. We also show that the order of the entire network can be extended while maintaining the structure of given functions.