The advent of data science has spurred interest in estimating properties of discrete distributions over large alphabets. Fundamental symmetric properties such as support size, support coverage, entropy, and proximity to uniformity, received most attention, with each property estimated using a different technique and often intricate analysis tools.

Motivated by the ... more >>>

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem arises in many diverse contexts ranging from tensor and operator norms to graph expansion to quantum information ... more >>>

We show that the total space in resolution, as well as in any other reasonable
proof system, is equal (up to a polynomial and $(\log n)^{O(1)}$ factors) to
the minimum refutation depth. In particular, all these variants of total space
are equivalent in this sense. The same conclusion holds for ... more >>>