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 0 . 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, Håstad 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. arkadev.c@tifr.res.in † Supported by a DAE fellowship. nikhil.mande@tifr.res.in
Chattopadhyay and Mande REF showed an exponential size separation between THR • MAJ and THR • THR, by constructing a function in THR • THR with exponential sign-rank.
Weights at the Bottom Matter When the Top is Heavy
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Mathematics Computer Science