Trevisan and Vadhan [TV, FOCS'00] constructed (seedless) extractors for distributions samplable by poly-size circuits that have very high min-entropy, under the very strong complexity theoretic hardness assumption that $\text{E}=\text{DTIME}(2^{O(n)})$ requires size $2^{\Omega(n)}$ circuits that have oracle access to $\Sigma_6^{\text{P}}$. In this paper we give three new explicit constructions of extractors ... more >>>

Given a joint distribution $P_{XY}$ of random variables $X,Y$ on finite alphabets, the information
bottleneck method seeks a representation $T$ of $X$ that is as informative as possible about $Y$ while compressing $X$; the privacy funnel asks for a representation that discloses much about $X$ while leaking little about $Y$. ... more >>>

We construct an explicit family of total sign matrices whose sign-rank is exponentially larger than their margin complexity. Previously, no superpolynomial separation was known for total matrices. In particular, we give a quantitative answer to a question of Linial, Mendelson, Schechtman, and Shraibman (Combinatorica, 2007), who asked whether the logarithmic ... more >>>