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info.json
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{
"abstract": "The principal submatrix localization problem deals with recovering a $K\\times K$ principal submatrix of elevated mean $\\mu$ in a large $n\\times n$ symmetric matrix subject to additive standard Gaussian noise, or more generally, mean zero, variance one, subgaussian noise. This problem serves as a prototypical example for community detection, in which the community corresponds to the support of the submatrix. The main result of this paper is that in the regime $\\Omega(\\sqrt{n}) \\leq K \\leq o(n)$, the support of the submatrix can be weakly recovered (with $o(K)$ misclassification errors on average) by an optimized message passing algorithm if $\\lambda = \\mu^2K^2/n$, the signal-to-noise ratio, exceeds $1/e$. This extends a result by Deshpande and Montanari previously obtained for $K=\\Theta(\\sqrt{n})$ and $\\mu=\\Theta(1).$ In addition, the algorithm can be combined with a voting procedure to achieve the information-theoretic limit of exact recovery with sharp constants for all $K \\geq \\frac{n}{\\log n} (\\frac{1}{8e} + o(1))$. The total running time of the algorithm is $O(n^2\\log n)$. Another version of the submatrix localization problem, known as noisy biclustering, aims to recover a $K_1\\times K_2$ submatrix of elevated mean $\\mu$ in a large $n_1\\times n_2$ Gaussian matrix. The optimized message passing algorithm and its analysis are adapted to the bicluster problem assuming $\\Omega(\\sqrt{n_i}) \\leq K_i \\leq o(n_i)$ and $K_1\\asymp K_2.$ A sharp information-theoretic condition for the weak recovery of both clusters is also identified.",
"authors": [
"Bruce Hajek",
"Yihong Wu",
"Jiaming Xu"
],
"id": "17-297",
"issue": 186,
"pages": [
1,
52
],
"title": "Submatrix localization via message passing",
"volume": 18,
"year": 2018
}