Numerical Tolerance for Spectral Decompositions of Random Matrices and Applications to Network Inference. Issue 1 (2nd January 2023) - Athreya, Avanti; Lubberts, Zachary; Priebe, Carey E.; Park, Youngser; Tang, Minh; Lyzinski, Vince; Kane, Michael; Lewis, Bryan W.
Abstract: We precisely quantify the impact of statistical error in the quality of a numerical approximation to a random matrix eigendecomposition, and under mild conditions, we use this to introduce an optimal numerical tolerance for residual error in spectral decompositions of random matrices. We demonstrate that terminating an eigendecomposition algorithm when the numerical error and statistical error are of the same order results in computational savings with no loss of accuracy. We illustrate the practical consequences of our stopping criterion with an analysis of simulated and real networks. Our theoretical results and real-data examples establish that the tradeoff between statistical and numerical error is of significant importance for subsequent inference.