The benefit of using random matrix theory to fit high-dimensional t-copulas

2016 ◽  
Author(s):  
Jiali Xu ◽  
Loïc Brin
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
High Dimensional

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.

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.

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory
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