A Sharp Lower Bound for Mixed-membership Estimation

Abstract

Consider an undirected network with $n$ nodes and $K$ perceivable communities, where some nodes may have mixed memberships. We assume that for each node $1 \leq i \leq n$, there is a probability mass function $\pi_i$ defined over $\{1, 2, \ldots, K\}$ such that \[ \pi_i(k) = \mbox{the weight of node $i$ on community $k$}, \qquad 1 \leq k \leq K. \] The goal is to estimate $\{\pi_i, 1 \leq i \leq n\}$ (i.e., membership estimation). We model the network with the {\it degree-corrected mixed membership (DCMM)} model \cite{Mixed-SCORE}. Since for many natural networks, the degrees have an approximate power-law tail, we allow {\it severe degree heterogeneity} in our model. For any membership estimation $\{\hat{\pi}_i, 1 \leq i \leq n\}$, since each $\pi_i$ is a probability mass function, it is natural to measure the errors by the average $\ell^1$-norm \[ \frac{1}{n} \sum_{i = 1}^n \| \hat{\pi}_i - \pi_i\|_1. \] We also consider a variant of the $\ell^1$-loss, where each $\|\hat{\pi}_i - \pi_i\|_1$ is re-weighted by the degree parameter $\theta_i$ in DCMM (to be introduced). We present a sharp lower bound. We also show that such a lower bound is achievable under a broad situation. More discussion in this vein is continued in our forthcoming manuscript. The results are very different from those on community detection. For community detection, the focus is on the special case where all $\pi_i$ are degenerate; the goal is clustering, so Hamming distance is the natural choice of loss function, and the rate can be exponentially fast. The setting here is broader and more difficult: it is more natural to use the $\ell^1$-loss, and the rate is only polynomially fast.