An efficiency upper bound for inverse covariance estimation

@article{Eldan2015AnEU,
  title={An efficiency upper bound for inverse covariance estimation},
  author={Ronen Eldan},
  journal={Israel Journal of Mathematics},
  year={2015},
  volume={207},
  pages={1-9}
}
  • Ronen Eldan
  • Published 3 December 2011
  • Mathematics
  • Israel Journal of Mathematics
We derive a quantitative upper bound for the efficiency of estimating entries in the inverse covariance matrix of a high dimensional distribution. We show that in order to approximate an off-diagonal entry of the density matrix of a d-dimensional Gaussian random vector, one needs at least a number of samples proportional to d. Furthermore, we show that with n ≪ d samples, the hypothesis that two given coordinates are fully correlated, when all other coordinates are conditioned to be zero… 
Testing for high‐dimensional geometry in random graphs
TLDR
The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix and a conjecture for the optimal detection boundary is made.
Entropic CLT and phase transition in high-dimensional Wishart matrices
TLDR
An information theoretic phase transition is proved: high dimensional Wishart matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if d is much larger than $n^3$ and the chain rule for relative entropy is used.
Random Geometric Graph: Some recent developments and perspectives
TLDR
This paper surveys the recent developments in RGGs from the lens of high dimensional settings and non-parametric inference and explains how this model differs from classical community based random graph models.
Commensurability of groups quasi-isometric to RAAGs
AbstractLet G be a right-angled Artin group with defining graph $$\Gamma $$Γ and let H be a finitely generated group quasi-isometric to G. We show if G satisfies that (1) its outer automorphism group
Basic models and questions in statistical network analysis
TLDR
This minicourse will investigate the most natural statistical questions for three canonical probabilistic models of networks: community detection in the stochastic block model, finding the embedding of a random geometric graph, and finding the original vertex in a preferential attachment tree.

References

SHOWING 1-8 OF 8 REFERENCES
Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles
Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with
Sparse permutation invariant covariance estimation
TLDR
A method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings using a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty is proposed.
Moments of minors of Wishart matrices
For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order m is populated by all m x
THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION
Br JOHN WISHART, M.A., B.Sc. Statistical Department, Rothamsted Experimental Station.
Introduction to the non-asymptotic analysis of random matrices
TLDR
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurementMatrices in compressed sensing.
Partial estimation of covariance matrices
TLDR
It is shown that a sample of size n = O(m log6p) is sufficient to accurately estimate in operator norm an arbitrary symmetric part of Σ consisting of m ≤ n nonzero entries per row.
Partial estimation of covariance matrices. Probability theory and related fields
  • 2009