We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC^0 circuits
for division, matrix powering, and related problems, where the improvement is in terms of ... more >>>

Given i.i.d. samples from an unknown distribution over a large domain $[N]$, approximating several basic quantities, including the distribution's support size, its entropy, and its distance from the uniform distribution, requires $\Theta(N / \log N)$ samples [Valiant and Valiant, STOC 2011].

Suppose, however, that we can interact with a powerful ... more >>>

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the distribution-free testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast ... more >>>