A sharp NMF result with applications in network modeling

Jiashun Jin

Keywords: Non-negative matrix factorization (NMF), social networks, degree-corrected block model, mixed-membership, low-rank model

TL;DR: We develop a sharp NMF result and use it to argue that in the most interesting regime of network analysis, we can rewrite a rank-K network model as a (recent) degree-corrected membership model.

Abstract:

Given an $n \times n$ non-negative rank-$K$ matrix $\Omega$ where $m$ eigenvalues are negative, when can we write $\Omega = Z P Z'$ for non-negative matrices $Z \in \mathbb{R}^{n, K}$ and $P \in \mathbb{R}^{K, K}$? While most existing works focused on the case of $m = 0$, our primary interest is on the case of general $m$. With new proof ideas we develop, we present sharp results on when the NMF problem is solvable, which significantly extend existing results on this topic. The NMF problem is partially motivated by applications in network modeling. For a network with $K$ communities, rank-$K$ models are popular, with many proposals. The DCMM model is
a recent rank-$K$ model which is especially useful and interpretable in practice. To enjoy such properties, it is of interest to study
when a rank-$K$ model can be rewritten as a DCMM model. Using our NMF results, we show that for a rank-$K$ model with parameters in the most interesting range, we can always rewrite it as a DCMM model.