We construct an explicit polynomial $f(x_1,...,x_n)$, with
coefficients in ${0,1}$, such that the size of any syntactically
multilinear arithmetic circuit computing $f$ is at least
$\Omega( n^{4/3} / log^2(n) )$. The lower bound holds over any field.

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We address well-studied problems concerning the learnability of parities and halfspaces in the presence of classification noise.

Learning of parities under the uniform distribution with random classification noise,also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding ... more >>>

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in AC^0[mod 2]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC^1. For example, we obtain the following results:

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