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 $mathbf{M}inmathbb{Z}^{mtimes m}$,
we want to find $mathbf{W}in{0,1}^{mtimes r}$ 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.

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Author Of this post: <a href="">Sitan Chen</a>, <a href="">Zhao Song</a>, <a href="">Runzhou Tao</a>, <a href="">Ruizhe Zhang</a>

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