Tail Bounds for Matrix Quadratic Forms and Bias Adjusted Spectral Clustering in Multi-layer Stochastic Block Models

03/18/2020

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by   Jing Lei, et al.

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We develop tail probability bounds for matrix linear combinations with matrix-valued coefficients and matrix-valued quadratic forms. These results extend well-known scalar case results such as the Hanson–Wright inequality, and matrix concentration inequalities such as the matrix Bernstein inequality. A key intermediate result is a deviation bound for matrix-valued U-statistics of order two and their independent sums. As an application of these probability tools in statistical inference, we establish the consistency of a novel bias-adjusted spectral clustering method in multi-layer stochastic block models with general signal structures.