Weights at the Bottom Matter When the Top is Heavy

Proving super-polynomial lower bounds against depth-2 threshold circuits of the form THR ◦THR is a well-known open problem that represents a frontier of our understanding in boolean circuit complexity. By contrast, exponential lower bounds on the size of THR◦MAJ circuits were shown by Razborov and Sherstov [31] even for computing functions in depth-3 AC. Yet, no separation among the two depth-2 threshold circuit classes were known. In fact, it is not clear a priori that they ought to be different. In particular, Goldmann, H̊astad and Razborov [14] showed that the class MAJ ◦MAJ is identical to the class MAJ ◦ THR. In this work, we provide an exponential separation between THR ◦MAJ and THR ◦ THR. We achieve this by showing a function f that is computed by linear size THR◦THR circuits and yet has exponentially large sign rank. This, by a well-known result, implies that f requires exponentially large THR ◦ MAJ circuits to be computed. Our result suggests that the sign rank method alone is unlikely to prove strong lower bounds against THR ◦ THR circuits. The main technical ingredient of our work is to prove a strong sign rank lower bound for an XOR function. This requires novel use of approximation theoretic tools. Partially supported by a Ramanujan fellowship of the DST. [email protected] Supported by a DAE fellowship. [email protected]