Electronic Journal of Statistics

The Kato–Temple inequality and eigenvalue concentration with applications to graph inference

Joshua Cape, Minh Tang, and Carey E. Priebe

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We present an adaptation of the Kato–Temple inequality for bounding perturbations of eigenvalues with applications to statistical inference for random graphs, specifically hypothesis testing and change-point detection. We obtain explicit high-probability bounds for the individual distances between certain signal eigenvalues of a graph’s adjacency matrix and the corresponding eigenvalues of the model’s edge probability matrix, even when the latter eigenvalues have multiplicity. Our results extend more broadly to the perturbation of singular values in the presence of quite general random matrix noise.

Article information

Electron. J. Statist., Volume 11, Number 2 (2017), 3954-3978.

Received: August 2016
First available in Project Euclid: 18 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions 15A42: Inequalities involving eigenvalues and eigenvectors
Secondary: 05C80: Random graphs [See also 60B20] 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]

Kato–Temple inequality eigenvalue concentration statistical inference for graphs perturbation theory random matrices

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Cape, Joshua; Tang, Minh; Priebe, Carey E. The Kato–Temple inequality and eigenvalue concentration with applications to graph inference. Electron. J. Statist. 11 (2017), no. 2, 3954--3978. doi:10.1214/17-EJS1328. https://projecteuclid.org/euclid.ejs/1508292532

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