Achieving the Bayes Error Rate in Synchronization and Block Models by SDP, Robustly

Abstract

We study the statistical performance of semidefinite programming (SDP) relaxations for clustering under random graph models. Under the Z 2 Synchronization model, Censored Block Model and Stochastic Block Model, we show that SDP achieves an error rate of the formHeren is an appropriate multiple of the number of nodes and I * is an information-theoretic measure of the signal-to-noise ratio. We provide matching lower bounds on the Bayes error for each model and therefore demonstrate that the SDP approach is Bayes optimal. As a corollary, our results imply that SDP achieves the optimal exact recovery threshold under each model. Furthermore, we show that SDP is robust: the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary. Our proof is based on a novel primal-dual analysis of SDP under a unified framework for all three models, and the analysis shows that SDP tightly approximates a joint majority voting procedure.

SDP Achieves Exact Minimax Optimality in Phase Synchronization

“…Despite being a continuous optimization method, the SDP is able to adapt to the discreteness of the problem. The exponential rate (11) has been previously derived for p = 1 by [13]. Our analysis based on the iterative algorithm perspective generalizes their result to more general values of p ≫ log n n .…”

“…The goal of our paper is to establish the statistical optimality of the SDP (13). We first provide a minimax lower bound as the benchmark of the problem.…”

“…The SDP can be extended to this more general setting by replacing all Y jk 's with A jk Y jk 's in (5). The formula will be given by (13) in Section 2.…”

“…Assume σ 2 = o(np) and np log n → ∞. Let Z be a global maximizer of the SDP (13) and z j = u j /|u j | for j ∈ [n] with u ∈ C n being the leading eigenvector of Z. There exists some δ = o(1) such that…”

Optimal Orthogonal Group Synchronization and Rotation Group Synchronization

“…Our paper is focused on the setting where d does not grow with the sample size n. This covers the most interesting applications in the literature for d = 3, though an extension of our result to a growing d would also be theoretically interesting. We exclude the case d = 1, because SO(1) is a degenerate set, and the problem over O(1) = {−1, 1} is known as Z 2 synchronization, whose minimax rate has already been derived in the literature [11,15]. It is interesting to note that the minimax rate of Z 2 synchronization is exponential instead of the polynomial rate of O(d) synchronization for d ≥ 2.…”