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.
翻译:在此工作中, 我们研究一种非负矩阵因子化的变体, 即我们希望找到一个给定输入矩阵的对称因子化为稀有的 Boulean 矩阵。 正式地说, 给$\ mathbf{M\ in\ mathb ⁇ m\ m} 美元, 我们想要找到 $\ mathbf{ 0. 1\m\ timer}, 例如 $\\ mathb{ m} - mathbf{ { w\ mathbf{ { mathf} - wequal- litude ral $@%0} 在所有的 mathbral 矩阵中最小化为美元。 这个问题与从线性图形中恢复超度的一些问题密切相关, 以及私人神经网络培训的重建攻击。 由于这个问题在最坏的情况下, 我们研究一种自然平均变体变体变体, 在这些重建攻击中, 很快会产生: $\\\\ fm} 我们的直线 roqreal deal rial deal deal deal deal rial deal rial deal deal $_ deal deal max_ amax amax max amax amax amax amax a has a bromax a bromax a m s b b b b b b b b b b b a ms a mex a ms a m sal a ms a max a ms a romotial a ms a ms a ms a ms a rial a motial a max a rial a max a max a max a max a ms a m sal a moal a mod a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a rial a ma a mod mod mod mod mod rial a f rial a rial a f a