Lower bounds for linear decision trees via an energy complexity argument

A linear decision tree is a binary decision tree in which a classification rule at each internal node is defined by a linear threshold function. In this paper, we consider a linear decision tree T where the weights w₁, w₂, ..., w_n of each linear threshold function satisfy ∑_i|w_i| ≤ w for an integer w, and prove that if T computes an n-variable Boolean function of large unbounded-error communication complexity (such as the Inner-Product function modulo two), then T must have 2^Ω(√n)/w leaves. To obtain the lower bound, we utilize a close relationship between the size of linear decision trees and the energy complexity of threshold circuits; the energy of a threshold circuit C is defined to be the maximum number of gates outputting "1," where the maximum is taken over all inputs to C. In addition, we consider threshold circuits of depth ω(1) and bounded energy, and provide two exponential lower bounds on the size (i.e., the number of gates) of such circuits.