We study depth-$4$ algebraic circuits with bounded top and bottom fan-in over fields of characteristic $0$. These circuits compute sums of at most $k$ products of polynomials of degree at most $\delta$. We denote this class of circuits by $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$. Our hard polynomial for this class is $P_{k + 1,n}$, ... more >>>

A boolean function $f$ has Fourier dimension $k$ if its strings with nonzero Fourier coefficients span a subspace of dimension $k$. We first consider the property testing task of determining whether a function has Fourier dimension at most $k$, or is $\epsilon$-far from being so. We show that there is ... more >>>

We give fast algorithms for low-rank tensor decomposition.
Over $\mathbb R$ and $\mathbb C$ we give a deterministic algorithm that takes as input an order-$d$, width-$n$, rank-$k$ tensor, gets measurement access to the tensor, runs in time
$k^{O(k^2)}\cdot \mathrm{poly}(n,d)$, and finds a minimal decomposition as a sum of rank-one ... more >>>