We give an explicit function $h:\{0,1\}^n\to\{0,1\}$ such that any deMorgan formula of size $O(n^{2.499})$ agrees with $h$ on at most $\frac{1}{2} + \epsilon$ fraction of the inputs, where $\epsilon$ is exponentially small (i.e. $\epsilon = 2^{-n^{\Omega(1)}}$). Previous lower bounds for formula size were obtained for exact computation.

The same ... more >>>

We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every deMorgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013).

Our technical contributions include a theorem that shows that the ``expected ... more >>>

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors.

1. Correlation bounds : We show that a random $d$-linear ... more >>>

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over $F_2$. We introduce a new technique to prove such correlation bounds with $F_2$ polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In ... more >>>

In this work we prove that there is a function $f \in \textrm{E}^\textrm{NP}$ such that, for every sufficiently large $n$ and $d = \sqrt{n}/\log n$, $f_n$ ($f$ restricted to $n$-bit inputs) cannot be $(1/2 + 2^{-d})$-approximated by $\textrm{F}_2$-polynomials of degree $d$. We also observe that a minor improvement ... more >>>