
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted cl...
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How Robust are Reconstruction Thresholds for Community Detection?
The stochastic block model is one of the oldest and most ubiquitous mode...
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Phase retrieval in high dimensions: Statistical and computational phase transitions
We consider the phase retrieval problem of reconstructing a ndimensiona...
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Tensor Clustering with Planted Structures: Statistical Optimality and Computational Limits
This paper studies the statistical and computational limits of highorde...
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Typology of phase transitions in Bayesian inference problems
Many inference problems, notably the stochastic block model (SBM) that g...
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Bayesian estimation from few samples: community detection and related problems
We propose an efficient metaalgorithm for Bayesian estimation problems ...
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Density Evolution in the Degreecorrelated Stochastic Block Model
There is a recent surge of interest in identifying the sharp recovery th...
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Computational phase transitions in sparse planted problems?
In recent times the cavity method, a statistical physicsinspired heuristic, has been successful in conjecturing computational thresholds that have been rigorously confirmed – such as for community detection in the sparse regime of the stochastic block model. Inspired by this, we investigate the predictions made by the cavity method for the algorithmic problems of detecting and recovering a planted signal in a general model of sparse random graphs. The model we study generalizes the wellunderstood case of the stochastic block model, the less well understood case of random constraint satisfaction problems with planted assignments, as well as "semisupervised" variants of these models. Our results include: (i) a conjecture about a precise criterion for when the problems of detection and recovery should be algorithmically tractable arising from a heuristic analysis of when a particular fixed point of the belief propagation algorithm is stable; (ii) a rigorous polynomialtime algorithm for the problem of detection: distinguishing a graph with a planted signal from one without; (iii) a rigorous polynomialtime algorithm for the problem of recovery: outputting a vector that correlates with the planted signal significantly better than a random guess would. The rigorous algorithms are based on the spectra of matrices that arise as the derivatives of the belief propagation update rule. An interesting unanswered question raised is that of obtaining evidence of computational hardness for convex relaxations whenever hardness is predicted by the cavity method.
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