  In this work, we study a variant of nonnegative matrix factorization where we
wish to find a symmetric factorization of a given input matrix into a sparse,
Boolean matrix. Formally speaking, given $\mathbf{M}\in\mathbb{Z}^{m\times m}$,
we want to find $\mathbf{W}\in\{0,1\}^{m\times r}$ such that $\| \mathbf{M} -
\mathbf{W}\mathbf{W}^\top \|_0$ is minimized among all $\mathbf{W}$ for which
each row is $k$-sparse. This question turns out to be closely related to a
number of questions like recovering a hypergraph from its line graph, as well
as reconstruction attacks for private neural network training.
  As this problem is hard in the worst-case, we study a natural average-case
variant that arises in the context of these reconstruction attacks: $\mathbf{M}
= \mathbf{W}\mathbf{W}^{\top}$ for $\mathbf{W}$ a random Boolean matrix with
$k$-sparse rows, and the goal is to recover $\mathbf{W}$ up to column
permutation. Equivalently, this can be thought of as recovering a uniformly
random $k$-uniform hypergraph from its line graph.
  Our main result is a polynomial-time algorithm for this problem based on
bootstrapping higher-order information about $\mathbf{W}$ and then decomposing
an appropriate tensor. The key ingredient in our analysis, which may be of
independent interest, is to show that such a matrix $\mathbf{W}$ has full
column rank with high probability as soon as $m = \widetilde{\Omega}(r)$, which
we do using tools from Littlewood-Offord theory and estimates for binary
Krawtchouk polynomials.
