Enumeration of linear threshold functions from the lattice of hyperplane intersections

Abstract

We present a method for enumerating linear threshold functions of n-dimensional binary inputs. Our starting point is the geometric lattice Ln of hyperplane intersections in the dual (weight) space. We show how the hyperoctahedral group O(n+1), the symmetry group of the (n+1)-dimensional hypercube, can be used to construct a symmetry-adapted poset of hyperplane intersections Deltan which is much more compact and tractable than Ln. A generalized Zeta function and its inverse, the generalized Möbius function, are defined on Deltan. Symmetry-adapted posets of hyperplane intersections for three-, four-, and five-dimensional inputs are constructed and the number of linear threshold functions is computed from the generalized Möbius function. Finally, we show how equivalence classes of linear threshold functions are enumerated by unfolding the symmetry-adapted poset of hyperplane intersections into a symmetry-adapted face poset. It is hoped that our construction will lead to ways of placing asymptotic bounds on the number of equivalence classes of linear threshold functions.