A Sharp Lower Bound for Mixed-membership Estimation

الأصل · EN

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 ≤ i ≤ n, there is a probability mass function πᵢ defined over {1, 2,, K} such that πᵢ(k) = the weight of node i on community k, 1 ≤ k ≤ K. The goal is to estimate {πᵢ, 1 ≤ i ≤ n} (i.e., membership estimation). We model the network with the degree-corrected mixed membership (DCMM) model Mixed-SCORE. Since for many natural networks, the degrees have an approximate power-law tail, we allow severe degree heterogeneity in our model. For any membership estimation {πᵢ, 1 ≤ i ≤ n}, since each πᵢ is a probability mass function, it is natural to measure the errors by the average ℓ¹-norm 1/n ∑ᵢ ₌ ₁ⁿ πᵢ - πᵢ₁. We also consider a variant of the ℓ¹-loss, where each πᵢ - πᵢ₁ is re-weighted by the degree parameter θᵢ 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 πᵢ 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 ℓ¹-loss, and the rate is only polynomially fast.