Non-asymptotic rate for high-dimensional covariance estimation with non-independent missing observations

2019 ◽  
Vol 153 ◽  
pp. 113-123
Author(s):  
Seongoh Park ◽  
Johan Lim
Keyword(s):  
Covariance Estimation ◽  
High Dimensional ◽  
Missing Observations ◽  
Asymptotic Rate

scholarly journals High-dimensional covariance estimation under the presence of outliers

2016 ◽  
Vol 9 (4) ◽  
pp. 461-468
Author(s):  
Hsin-Cheng Huang ◽  
Thomas C. M. Lee
Keyword(s):  
Covariance Estimation ◽  
High Dimensional

scholarly journals High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence

2011 ◽  
Vol 5 (0) ◽  
pp. 935-980 ◽  
Author(s):  
Pradeep Ravikumar ◽  
Martin J. Wainwright ◽  
Garvesh Raskutti ◽  
Bin Yu
Keyword(s):  
Covariance Estimation ◽  
High Dimensional

Convergence rates of high dimensional Smolyak quadrature

2020 ◽  
Vol 54 (4) ◽  
pp. 1259-1307
Author(s):  
Jakob Zech ◽  
Christoph Schwab
Keyword(s):  
Convergence Rates ◽  
A Priori ◽  
Computational Cost ◽  
High Dimensional ◽  
Taylor Polynomial ◽  
Approximation Rates ◽  
Moderate Number ◽  
Asymptotic Rate ◽  
Smolyak Algorithm ◽  
Parametric Maps

We analyse convergence rates of Smolyak integration for parametric maps u: U → X taking values in a Banach space X, defined on the parameter domain U = [−1,1]N. For parametric maps which are sparse, as quantified by summability of their Taylor polynomial chaos coefficients, dimension-independent convergence rates superior to N-term approximation rates under the same sparsity are achievable. We propose a concrete Smolyak algorithm to a priori identify integrand-adapted sets of active multiindices (and thereby unisolvent sparse grids of quadrature points) via upper bounds for the integrands’ Taylor gpc coefficients. For so-called “(b,ε)-holomorphic” integrands u with b∈lp(∕) for some p ∈ (0, 1), we prove the dimension-independent convergence rate 2/p − 1 in terms of the number of quadrature points. The proposed Smolyak algorithm is proved to yield (essentially) the same rate in terms of the total computational cost for both nested and non-nested univariate quadrature points. Numerical experiments and a mathematical sparsity analysis accounting for cancellations in quadratures and in the combination formula demonstrate that the asymptotic rate 2/p − 1 is realized computationally for a moderate number of quadrature points under certain circumstances. By a refined analysis of model integrand classes we show that a generally large preasymptotic range otherwise precludes reaching the asymptotic rate 2/p − 1 for practically relevant numbers of quadrature points.

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