Sphericity and Identity Test for High-dimensional Covariance Matrix Using Random Matrix Theory

2021 ◽  
Vol 37 (2) ◽  
pp. 214-231
Author(s):  
Shou-cheng Yuan ◽  
Jie Zhou ◽  
Jian-xin Pan ◽  
Jie-qiong Shen
Keyword(s):  
Covariance Matrix ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
High Dimensional ◽  
Identity Test

scholarly journals Sparse recovery from extreme eigenvalues deviation inequalities

2019 ◽  
Vol 23 ◽  
pp. 430-463
Author(s):  
Sandrine Dallaporta ◽  
Yohann De Castro
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
State Of The Art ◽  
Singular Values ◽  
Upper Bounds ◽  
Sparse Recovery ◽  
High Dimensional ◽  
Extreme Eigenvalues ◽  
Explicit Derivation

This article provides a new toolbox to derive sparse recovery guarantees – that is referred to as “stable and robust sparse regression” (SRSR) – from deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in SRSR. While it is an open problem to construct deterministic matrices with apposite RICs, one can prove that such matrices exist using random matrices models. In this paper, we show upper bounds on RICs for Gaussian and Rademacher matrices using state-of-the-art deviation estimates on their extreme eigenvalues. This allows us to derive a lower bound on the probability of getting SRSR. One benefit of this paper is a direct and explicit derivation of upper bounds on RICs and lower bounds on SRSR from deviations on the extreme eigenvalues given by Random Matrix theory.

scholarly journals Spectrum Sensing in Cognitive Radio by Use of Volume-Based Method

2018 ◽  
Vol 7 (2.17) ◽  
pp. 34
Author(s):  
C S. Preetham ◽  
Ch Mahesh ◽  
Ch Saranga Haripriya ◽  
Ramaraju Anirudh ◽  
M S. Sireesha
Keyword(s):  
Covariance Matrix ◽  
Spectrum Sensing ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Sample Covariance Matrix ◽  
Closed Form Expression ◽  
Sample Covariance ◽  
Signal Sample ◽  
Deficiency Cases

Spectrum sensing is the mission of finding the licensed user signal situation, i.e. to determine the existence and deficiency of primary (licensed) user signal, the recent publications random matrix theory algorithms performs better-quality in spectrum sensing. The RMT fundamental nature is to make use of the distributed extremal eigenvalues of the arrived signal sample covariance matrix (SMC), specifically, Tracy-Widom (TW) distribution which is useful to certain extent in spectrum sensing but demanding for numerical evaluations because there is absence of closed-form expression in it. The sample covariance matrix determinant is designed for two novel volume-based detectors or signal existence and deficiency cases are differentiated by using volume. Under the Gaussian noise postulation one of the detectors theoretical decision thresholds is perfectly calculated by using Random matrix theory. The volume-based detectors efficiency is shown in simulation results. 

Estimation of error variance via ridge regression

Biometrika ◽  
2020 ◽  
Author(s):  
X Liu ◽  
S Zheng ◽  
X Feng
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Ridge Regression ◽  
Matrix Theory ◽  
Numerical Study ◽  
Asymptotic Properties ◽  
Error Variance ◽  
High Dimensional ◽  
Finite Sample ◽  
Estimation Of Error

Summary We propose a novel estimator of error variance and establish its asymptotic properties based on ridge regression and random matrix theory. The proposed estimator is valid under both low- and high-dimensional models, and performs well not only in nonsparse cases, but also in sparse ones. The finite-sample performance of the proposed method is assessed through an intensive numerical study, which indicates that the method is promising compared with its competitors in many interesting scenarios.

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