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 >>>

Given a Boolean circuit $C$, we wish to convert it to a circuit $C'$ that computes the same function as $C$ even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [KLM97]. Can we ... more >>>