Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,\Sigma)$, where each element $x \in X$ lies in $t$ randomly selected sets of $\Sigma$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We ... more >>>

We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions. Our main complexity-theoretic result proves that most linear program feasibility problems cannot be computed by polynomial-sized constant-depth circuits. Moreover, our result ... more >>>

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for ... more >>>