On the Size of Depth-Two Threshold Circuits for the Inner Product Mod 2 Function.

In this paper, we study the size of depth-two threshold circuits computing the inner product mod 2 function (mod 2). First, we reveal that can be computed by a depth-two threshold circuit of size significantly smaller than a folklore construction of size . Namely, we give a construction of such a circuit (denoted by circuit) of size . We also give an upper bound of for the case that the weights of the top threshold gate are polynomially bounded (denoted by circuit). Second, we give new lower bounds on the size of depth-two circuits of some special form; the top gate is an unbounded weight threshold gate and the bottom gates are symmetric gates (denoted by circuit). We show that any such circuit computing has size for every constant . This improves the previous bound of based on the sign-rank method due to Forster et al. [JCSS ’02, FSTTCS ’01]. Our technique has a unique feature that the lower bound is obtained by giving an explicit feasible solution to (the dual of) a certain linear programming problem. In fact, the problem itself was presented by the author over a decade ago [MFCS ’05], and finding a good solution is an actual contribution of this work.