In a broad Degree-Corrected Mixed-Membership (DCMM) setting, we test whether a non-uniform hypergraph has only one community or has multiple communities. Since both the null and alternative hypotheses have many unknown parameters, the challenge is, given an alternative, how to identify the null that is hardest to separate from the alternative. We approach this by proposing a degree matching strategy where the main idea is leveraging the theory for tensor scaling to create a least favorable pair of hypotheses. We present a result on standard minimax lower bound theory and a result on Region of Impossibility (which is more informative than the minimax lower bound). We show that our lower bounds are tight by introducing a new test that attains the lower bound up to a logarithmic factor. We also discuss the case where the hypergraphs may have mixed-memberships.

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number K of clusters of sizes linearly growing with the global population of items n. Every pair of items is labeled independently at random, and label ` appears with probability p(i,j,`) between two items in clusters indexed by iand j, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most s misclassified items in average under the general LSBM and for any s = o(n), which solves one open problem raised in [2]. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within O(npolylog(n)) computations and without the a-priori knowledge of the model parameters.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we propose a general method to solve the localization problem by learning a regularization matrix from the localized eigenvectors. Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors representing the global structure. We show applications of our method in several inference problems: community detection in networks, clustering from pairwise similarities, rank estimation and matrix completion problems. Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. This is in contrast with existing spectral algorithms based on data matrix, non-backtracking matrix, Laplacians and those with rank-one regularizations, which perform poorly in the sparse case with noise.

We consider a crowdsourcing model in which nworkers are asked to rate the quality of nitems previously generated by other workers. An unknown set of αnworkers generate reliable ratings, while the remaining workers may behave arbitrarily and possibly adversarially. The manager of the experiment can also manually evaluate the quality of a small number of items, and wishes to curate together almost all of the high-quality items with at most anfraction of low-quality items.

PageRank (PR), an algorithm originally proposed by Page et al. for ranking web-pages [1] has found manysuccessful applications, including community detection [2,3],linkprediction [4]and recommendersystemdesign[5,6].