Separating Points by Parallel Hyperplanes— Characterization Problem

This paper deals with partitions of a discrete set $S$ of points in a $d$-dimensional space, by $h$ parallel hyperplanes. Such partitions are in a direct correspondence with multilinear threshold functions which appear in the theory of neural networks and multivalued logic. The characterization (encoding) problem is studied. We show that a unique characterization (encoding) of such multilinear partitions of $S = \{0, 1, \ldots, m-1\}^d$ is possible within $ {\cal O}(h\cdot d^2\cdot \log m) $ bit rate per encoded partition. The proposed characterization (code) consists of $(d+1)\cdot (h+1)$ discrete moments having the order no bigger than 1. The obtained bit rate is evaluated depending on the mutual relations between $h, d$, and $m$. The optimality is reached in some cases.