This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector X ∈ R n with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of X itself. Our results recover, in particular, the classic uniform bound of Talagrand (1996) for Rademacher chaoses and the more recent uniform result of Adamczak (2015) which holds under certain rather strong assumptions on the distribution of X. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici (2014). Finally, we show a uniform Hanson-Wright-type inequality in the Ising model under Dobrushin's condition. A closely related question was posed by Marton (2003).

We consider the robust algorithms for the k-means clustering problem where a quantizer is constructed based on N independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space. In particular, our results extend the renowned asymptotic result of Pollard (1981) who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer in R d . In a special case of clustering in R d , under two bounded moments, we prove matching (up to constant factors) non-asymptotic upper and lower bounds on the excess distortion, which depend on the probability mass of the lightest cluster of an optimal quantizer. Our bounds have the sub-Gaussian form, and the proofs are based on the versions of uniform bounds for robust mean estimators.

We derive a Central Limit Theorem (CLT) for log det MN / √ N − 2θN , where MN is from the Gaussian Unitary or Gaussian Orthogonal Ensemble (GUE and GOE), and 2θN is local to the edge of the semicircle law. Precisely, 2θN = 2 + N −2/3 σN with σN being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that σN log 2 N . For slowly growing σN , our proofs hold for general Gaussian β-ensembles. We also extend our CLT to cover spiked GUE and GOE.

No abstract

No abstract