Low coordinate degree algorithms II: Categorical signals and generalized stochastic block models

Abstract: We study when low coordinate degree functions (LCDF) -- linear combinations of functions depending on small subsets of entries of a vector -- can test for the presence of categorical structure, including community structure and generalizations thereof, in high-dimensional data. This complements the first paper of this series, which studied the power of LCDF in testing for continuous structure like real-valued signals perturbed by additive noise. We apply the tools developed there to a general form of stochastic block model (SBM), where a population is assigned random labels and every $p$-tuple of the population generates an observation according to an arbitrary probability measure associated to the $p$ labels of its members. We show that the performance of LCDF admits a unified analysis for this class of models. As applications, we prove tight lower bounds against LCDF (and therefore also against low degree polynomials) for nearly arbitrary graph and regular hypergraph SBMs, always matching suitable generalizations of the Kesten-Stigum threshold. We also prove tight lower bounds for group synchronization and abelian group sumset problems under the "truth-or-Haar" noise model, and use our technical results to give an improved analysis of Gaussian multi-frequency group synchronization. In most of these models, for some parameter settings our lower bounds give new evidence for conjectural statistical-to-computational gaps. Finally, interpreting some of our findings, we propose a precise analogy between categorical and continuous signals: a general SBM as above behaves, in terms of the tradeoff between subexponential runtime cost of testing algorithms and the signal strength needed for a testing algorithm to succeed, like a spiked $p_$-tensor model of a certain order $p_$ that may be computed from the parameters of the SBM.