MIXANDMIX: numerical techniques for the computation of empirical spectral distributions of population mixtures

Lucilio Cordero-Grande

doi:10.1016/j.csda.2019.06.011

Lucilio Cordero-Grande

Open Access

https://doi.org/10.1016/j.csda.2019.06.011

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Abstract

The MIXANDMIX (mixtures by Anderson mixing) tool for the computation of the empirical spectral distribution of random matrices generated by mixtures of populations is described. Within the population mixture model the mapping between the population distributions and the limiting spectral distribution can be obtained by solving a set of systems of non-linear equations, for which an efficient implementation is provided. The contributions include a method for accelerated fixed point convergence, a homotopy continuation strategy to prevent convergence to non-admissible solutions, a blind non-uniform grid construction for effective distribution support detection and approximation, and a parallel computing architecture. Comparisons are performed with available packages for the single population case and with results obtained by simulation for the more general model implemented here. Results show competitive performance and improved flexibility.

Highlights

Random matrix theory is at the core of modern high dimensional statistical inference (Yao et al, 2015) with applications in physics, biology, economics, communications, computer science or imaging (Couillet and Debbah, 2013; Paul and Aue, 2014; Bun et al, 2017)
Silverstein and Choi (1995) develops ideas outlined in Marčenko and Pastur (1967) to analyse the support of the empirical spectral distribution (ESD) based on the monotonicity of the inverse of the function involved in the fixed point equation
The most flexible package that we have identified has been described in Dobriban (2015) (SPECTRODE), where the fixed point equation in Silverstein and Bai (1995) is transformed into an ordinary differential equation (ODE) with initial condition obtained by the solution of the fixed point equation on a single point within the support

Summary

Introduction

Random matrix theory is at the core of modern high dimensional statistical inference (Yao et al, 2015) with applications in physics, biology, economics, communications, computer science or imaging (Couillet and Debbah, 2013; Paul and Aue, 2014; Bun et al, 2017). A key contribution in this setting is the Marčenko–Pastur theorem (Marčenko and Pastur, 1967; Silverstein and Bai, 1995), which characterizes the limiting behaviour of the empirical spectral distribution (ESD) for matrices with random entries when N → ∞. There exists a fixed point equation relating the eigenvalues of the empirical and population distributions, which can be used for inference under the mediation of appropriate numerical techniques. Using this characterization, subsequent asymptotics-based inference can be performed, for instance, for dimensionality reduction, hypothesis testing, signal retrieval, classification or covariance estimation (Yao et al, 2015). The relation between the population covariances and the limiting behaviour of the sample eigenvalues is governed by a system of non-linear equations and, asymptotic sample eigenvalue confinement has been proved (Kammoun and Alouini, 2016), a simple description of the support is no longer at hand

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Discussion

Conclusion