RECORD
Covariance estimation for distributions with $2+\varepsilon$ moments
Probability · Submitted 14 Jun 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We establish the optimal bound N=O(n) for every distribution whose k-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound N=O(n) for every distribution whose k-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan-Lovasz-Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535-561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption - uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704-1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. | background 0.51 | 0.57 | 0.06 | unclear 0.66 |
| s1 | We establish the optimal bound N=O(n) for every distribution whose k-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. | result 0.99 | 0.95 | 0.87 | asserted 1.00 |
| s2 | In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan-Lovasz-Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535-561]. | interpretation 0.90 | 0.43 | 0.06 | asserted 0.87 |
| s3 | Moreover, a lower estimate on the covariance matrix holds under a weaker assumption - uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. | result 0.98 | 0.89 | 0.06 | asserted 1.00 |
| s4 | Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704-1721]. | method 0.98 | 0.76 | 0.03 | asserted 1.00 |
| s5 | The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points. | method 0.99 | 0.79 | 0.04 | asserted 0.86 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound N=O(n) for every distribution whose k-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan-Lovasz-Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535-561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption - uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704-1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.
Human review
How the literature uses it
6 arXiv papers cite this in 7 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 1 paper
-
Specifically, letting $κ_ℓ= λ_1(C)/λ_ℓ(C),$ we see that $O(ε^-2κ_ℓ^2 p)$ measurements are required to obtain relative-error estimates of the dominant $ℓ$ eigenvalues of $C$ using the results of [ref] .
Tail bounds for all eigenvalues of a sum of random matrices · Covariance Estimation · 0.88
Cites it as background 4 papers
-
The finite sample performance of the sample covariance matrix was also profoundly investigated for a large class of distributions, see [ref] and references therein.
Error Bound for Compound Wishart Matrices · Introduction · 0.86
-
The third kind of results stems from the non-asymptotic analysis of the SCM, see e.g. [ref] .
Performance Analysis of Tyler's Covariance Estimator · Introduction · 0.94
-
A broad exposition on sample covariance performance for a large class of distributions was performed in [ref] .
Tyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure · Sample Covariance · 0.90
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.