Covariance estimation for distributions with 2+ε moments

By Nikhil Srivastava, Roman Vershynin. 2011

128 citations recorded by the source corpus.

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+e moments outside the sphere of radius O(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+e 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.

Subfields: Artificial Intelligence, Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c6e39903e53613217778fb0e52aa1bc958c4327b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Ramon van Handel
2. Konstantin E. Tikhomirov
3. Xiaolu Tan
4. Marcel Nutz
5. Nicole El Karoui

In Computational Mechanics

1. Ramon van Handel
2. Konstantin E. Tikhomirov

In Statistics and Probability

1. Ramon van Handel
2. Konstantin E. Tikhomirov
3. Justin Salez
4. Xiaolu Tan
5. Marcel Nutz
6. Nicole El Karoui

Citations in the corpus

• The smallest singular value of random rectangular matrices with no moment assumptions on entries (2014, background): “in assumptions on the aspect ratio N/n. It is important to note that the conditions imposed on A are too weak to imply the “usual” upper bound s1(A) . √ N with a large probability [8]. This result of [17] was strengthened by V. Koltchinskii and S. Mendelson in [5] under similar assumptions on the ma”
• Structured Random Matrices (2016, methodology): “[23] where related ideas were applied to Stieltjes transforms in a random matrix problem).”