  In this paper, we investigate computational power of threshold circuits and
other theoretical models of neural networks in terms of the following four
complexity measures: size (the number of gates), depth, weight and energy. Here
the energy complexity of a circuit measures sparsity of their computation, and
is defined as the maximum number of gates outputting non-zero values taken over
all the input assignments. As our main result, we prove that any threshold
circuit $C$ of size $s$, depth $d$, energy $e$ and weight $w$ satisfies $\log
(rk(M_C)) \le ed (\log s + \log w + \log n)$, where $rk(M_C)$ is the rank of
the communication matrix $M_C$ of a $2n$-variable Boolean function that $C$
computes. Thus, such a threshold circuit $C$ is able to compute only a Boolean
function of which communication matrix has rank bounded by a product of
logarithmic factors of $s,w$ and linear factors of $d,e$. This implies an
exponential lower bound on the size of even sublinear-depth threshold circuit
if energy and weight are sufficiently small. For other models of neural
networks such as a discretized ReLE circuits and decretized sigmoid circuits,
we prove that a similar inequality also holds for a discretized circuit $C$:
$rk(M_C) = O(ed(\log s + \log w + \log n)^3)$.
