A note on the Hanson-Wright inequality for random vectors with dependencies

By Radosław Adamczak. 2014

128 citations recorded by the source corpus.

We prove that quadratic forms in isotropic random vectors $X$ in $\mathbb{R}^n$, possessing the convex concentration property with constant $K$, satisfy the Hanson-Wright inequality with constant $CK$, where $C$ is an absolute constant, thus eliminating the logarithmic (in the dimension) factors in a recent estimate by Vu and Wang. We also show that the concentration inequality for all Lipschitz functions implies a uniform version of the Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the inequalities by Borell, Arcones-Gine and Ledoux-Talagrand). Previous results of this type relied on stronger isoperimetric properties of $X$ and in some cases provided an upper bound on the deviations rather than a concentration inequality. In the last part of the paper we show that the uniform version of the Hanson-Wright inequality for Gaussian vectors can be used to recover a recent concentration inequality for empirical estimators of the covariance operator of $B$-valued Gaussian variables due to Koltchinskii and Lounici.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2b5e20eb724f8f6ad2a2d403f8d3d77e62a56e40

How researchers see this paper

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

In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Concentration inequalities for polynomials in α-sub-exponential random variables (2019, background): “The case of α ∈ [1, 2] and polynomials of total degree D ≤ 3 has been discussed in [4, Proposition 6.”
Non-asymptotic Identification of LTI Systems from a Single Trajectory (2018, methodology): “3 of [1], we ﬁnd the concentration bound P ( s a ≥ ( τ + 1 ) N t ∥ Γ ∞ γ ) ≤ 2exp (− 2 cτ min { which is the desired result after 1 + τ ↔ τ substitution and using the initial assumption of τ ≥ 2 .”
Learning Compact Neural Networks with Regularization (2018, result): “asymmetric version of main theorem of [1] yields (in a similar fashion to Corollary D.”
Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method (2018, background): “Similarly, one can introduce the moment analog of logarithmic Sobolev inequality (see equation 3 in Adamczak and Wolff (2015)): ‖Z(X) − EZ(X)‖Lq ≤ K √ q‖|∇Z(X)|‖Lq , where K > 0 is a constant, | · | stands for the standard Euclidian norm and q ≥ 2.”
Concentration inequalities and moment bounds for sample covariance operators (2014, methodology): “Adamczak [1] suggested an alternative approach to the proof of Theorem 5.”