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 M∈Zm×m,
we want to find W∈{0,1}m×r such that ∥M−WW⊤∥0 is minimized among all 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: M=WW⊤ for W a random Boolean matrix with
k-sparse rows, and the goal is to recover 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 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 W has full
column rank with high probability as soon as m=Ω(r), which
we do using tools from Littlewood-Offord theory and estimates for binary
Krawtchouk polynomials.Comment: 33 pages, to appear in Innovations in Theoretical Computer Science
(ITCS 2022), v2: updated ref