Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond

By Noureddine El Karoui. 2009

130 citations recorded by the source corpus.

We place ourselves in the setting of high-dimensional statistical inference, where the number of variables $p$ in a data set of interest is of the same order of magnitude as the number of observations $n$. More formally, we study the asymptotic properties of correlation and covariance matrices, in the setting where $p/n\to\rho\in(0,\infty),$ for general population covariance. We show that, for a large class of models studied in random matrix theory, spectral properties of large-dimensional correlation matrices are similar to those of large-dimensional covarance matrices. We also derive a Mar\u{c}enko--Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.

Subfields: Economics and Econometrics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6986e599ce0b874718f21cbf2547d205fe620f44

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Citations in the corpus

High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation (2010, methodology): “We give only in this introduction its simplest form and refer the reader to Mar cenko and Pastur (1967), El Karoui (2008b) and El Karoui (2009) for a more thorough introduction and very recent developments, as well as potential geometric and statistical limitations of the models usually considered”
On the Marčenko-Pastur law for linear time series (2013, background): “First, the class of time series models that are analyzed in detail encompasses the setting of stationary i.i.d. rows studied by Jin et al. [21], Pfaffel and Schlemm [32] and Yao [35], as well as the setting of separable covariance structure studied by Zhang [36], Paul and Silverstein [31] and El Kar”
On the impact of predictor geometry on the performance on high-dimensional ridge-regularized generalized robust regression estimators (2018, background): “Importantly, note thatO4 does not require the entries ofXi to be independent; see [27] or [13] for examples of Xi satisfyingO4 with dependent entries.”
On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries (2011, background): “313] in the case of matrices with independent entries, however (as noted in [14]) in fact just independence of rows is required.”
Capturing the learning curves of generic features maps for realistic data sets with a teacher-student model (2021, background): “While this is still restrictive, stronger results can be shown, and [59, 61, 76, 47] extended them (also loosening the independence assumption) for a very generic class of distributions of correlated random vectors u.”
Random matrix theory in statistics: A review (2014, methodology): “El Karoui (2009a), Paul and Silverstein (2009) and Zhang (2006) showed that the Stieltjes transform of the limiting ESD is obtained as the unique solution of a coupled functional equation.”
Cleaning large correlation matrices: tools from random matrix theory (2016, background): “3, it is possible to show that for the U(x) = x−1, one has M → E in the large N limit [100, 101, 105, 106], where E = CWC and W is a Wishart matrix.”
The spectrum of kernel random matrices (2010, methodology): “The proof of this last fact follows the same step as that of Lemma A-4, with a slight adjustment since we need to replace 2 by b. For a related question and more details, we refer the reader to El Karoui (2007a) .”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “The spectral properties of N−1YNY∗ N have been widely investigated in some recent work under the highdimensional setting, for example, one can refer to [13, 42, 57, 58].”
Asymptotic behavior of unregularized and ridge-regularized high-dimensional robust regression estimators : rigorous results (2013, background): “Once again, the approximating quantities we consider are “very natural” in light of our work in El Karoui et al. (2013).”
A Random Matrix Approach to Neural Networks (2017, methodology): “Here, to alleviate the non-linear difficulty, we exploit concentration of measure arguments (Ledoux, 2005), thereby pushing further the original ideas of (El Karoui, 2009) established for simpler random matrix models.”