Noga Alon, Shay Moran, Amir Yehudayoff

We study the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is $3$. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. Similar (slightly less accurate) statements hold for $d>2$ as well. We discuss the tightness of our methods, and describe ... more >>>

Arkadev Chattopadhyay, Nikhil Mande

Proving super-polynomial lower bounds against depth-2 threshold circuits of the form THR of 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 of MAJ circuits were shown by Razborov and Sherstov (SIAM ... more >>>