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 >>>

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 >>>