Feedforward neural networks with sigmoid activations are of fundamental importance in neural computation. However, our understanding of the circuit complexity of such networks for implementing Boolean functions remains limited. The analog output of the sigmoid, along with its transcendental nature, leads to unique mathematical challenges for proving lower bounds.

... more >>>

We show that over the field of rational numbers, there is a deterministic algorithm for factorization of constant depth circuits that makes calls to a (whitebox or blackbox) PIT algorithm for such circuits. The time complexity of the algorithm (and hence the number of calls to the PIT oracle) is ... more >>>

The best known lower bound on the determinantal complexity of the $4\times4$ permanent is 10. In this short note, I prove that this value is exactly 15: the $4\times4$ permanent is the determinant of a $15\times15$ matrix of affine linear forms, and cannot be that of any smaller matrix. To ... more >>>