Statistical and computational thresholds for the planted $k$-densest sub-hypergraph problem

Corinzia, Luca; Penna, P. La; Szpankowski, Wojciech; Buhmann, Joachim M.

doi:10.48550/arxiv.2011.11500

Cited by 2 publications

(3 citation statements)

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“…• Planted Densest Sub-hypergraph Model: Let 0 ≤ q < p ≤ 1 and r = 1. There is a densest subhypergraph planted in the hypergraph, which generalizes the sub-graph model in [7] and is somewhat similar to [10] except that they assume Gaussiandistributed weights.…”

Section: Remark 3 Our Settings Cover the Following Classical Models W...mentioning

confidence: 99%

“…[8,14,15] considered the asymptotic statistical lower bound instead of the more challenging finite sample information lower bound and did not model isolated nodes. [10] modelled isolated nodes and assumed that the hyperedges have Gaussian weights which is mathematically interesting, but not a typical assumption in practice, due to the lack of applications in the realworld for this regime. [2,3,9] focused on stochastic block models and did not model isolated nodes.…”

Section: Introductionmentioning

confidence: 99%

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Information Theoretic Limits of Exact Recovery in Sub-hypergraph Models for Community Detection

Liang¹,

Ke²,

Honorio³

2021

Preprint

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In this paper, we study the information theoretic bounds for exact recovery in sub-hypergraph models for community detection. We define a general model called the m−uniform sub-hypergraph stochastic block model (m−ShSBM). Under the m−ShSBM, we use Fano's inequality to identify the region of model parameters where any algorithm fails to exactly recover the planted communities with a large probability. We also identify the region where a Maximum Likelihood Estimation (MLE) algorithm succeeds to exactly recover the communities with high probability. Our bounds are tight and pertain to the community detection problems in various models such as the planted hypergraph stochastic block model, the planted densest sub-hypergraph model, and the planted multipartite hypergraph model.

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Section: Remark 3 Our Settings Cover the Following Classical Models W...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Information Theoretic Limits of Exact Recovery in Sub-hypergraph Models for Community Detection

Liang¹,

Ke²,

Honorio³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…However, to the best of our knowledge, only a few other works studied how the same AoN phenomenon extends to other estimators. Examples include [17], [18], where non-matching upper and lower bounds are provided for the transition of the vectorial-MLE in the sparse planted hypergraph problem (equivalent to sparse tensor-PCA up to a reparameterization of the dimensionality of the problem), and [19] where the AoN is proved in the sparse linear regression model for the vectorial-MLE estimator.…”

Section: B Related Workmentioning

confidence: 99%

On maximum-likelihood estimation in the all-or-nothing regime

Corinzia¹,

Penna²,

Szpankowski³

et al. 2021

Preprint

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We study the problem of estimating a rank-1 additive deformation of a Gaussian tensor according to the maximumlikelihood estimator (MLE). The analysis is carried out in the sparse setting, where the underlying signal has a support that scales sublinearly with the total number of dimensions. We show that for Bernoulli distributed signals, the MLE undergoes an all-or-nothing (AoN) phase transition, already established for the minimum mean-square-error estimator (MMSE) in the same problem. The result follows from two main technical points: (i) the connection established between the MLE and the MMSE, using the first and second-moment methods in the constrained signal space, (ii) a recovery regime for the MMSE stricter than the simple error vanishing characterization given in the standard AoN, that is here proved as a general result.

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Statistical and computational thresholds for the planted $k$-densest sub-hypergraph problem

Cited by 2 publications

References 0 publications

Information Theoretic Limits of Exact Recovery in Sub-hypergraph Models for Community Detection

Information Theoretic Limits of Exact Recovery in Sub-hypergraph Models for Community Detection

On maximum-likelihood estimation in the all-or-nothing regime

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