In this paper, we apply tools from algebraic geometry to prove new results concerning extractors for algebraic sets, the recursive Fourier sampling problem, and VC dimension. We present a new construction of an extractor which works for algebraic sets defined by polynomials over $\mathbb{F}_2$ of substantially higher degree than the ... more >>>

We prove that any algorithm for learning parities requires either a memory of quadratic size or an exponential number of samples. This proves a recent conjecture of Steinhardt, Valiant and Wager and shows that for some learning problems a large storage space is crucial.

More formally, in the problem of ... more >>>

We study two variants of seeded randomness extractors. The first one, as studied by Goldreich et al. \cite{goldreich2015randomness}, is seeded extractors that can be computed by $AC^0$ circuits. The second one, as introduced by Bogdanov and Guo \cite{bogdanov2013sparse}, is (strong) extractor families that consist of sparse transformations, i.e., functions that ... more >>>