Lower Bounds for Linear Decision Trees with Bounded Weights

Abstract

In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w + 1)^r threshold gates with weight at most (2w + 1)^r + 1 in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2^Ω(n/logw) lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound $2^{\sqrt{n}}$ if $w=2^{o(\sqrt{n})}$.