Hanson-Wright inequality and sub-gaussian concentration

In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables.We deduce a useful concentration inequality for sub-gaussian random vectors.Two examples are given to illustrate these results: a concentration of distances between random vectors and subspaces, and a bound on the norms of products of random and deterministic matrices.

[1]  C. James DEPENDENCE OR INDEPENDENCE. , 1963, Hospital management.

[2]  F. T. Wright,et al.  A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables , 1971 .

[3]  F. T. Wright A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables Whose Distributions are not Necessarily Symmetric , 1973 .

[4]  V. Peña,et al.  Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics , 1993, math/9309211.

[5]  M. Talagrand Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[6]  J. Bourgain Random Points in Isotropic Convex Sets , 1998 .

[7]  M. Ledoux The concentration of measure phenomenon , 2001 .

[8]  R. Latala Estimates of moments and tails of Gaussian chaoses , 2005, math/0505313.

[9]  R. Latala Estimates of moments and tails of Gaussian chaoses , 2005, math/0505313.

[10]  Krzysztof Oleszkiewicz,et al.  Banach-Mazur Distances and Projections on Random Subgaussian Polytopes , 2007, Discret. Comput. Geom..

[11]  R. Vershynin Spectral norm of products of random and deterministic matrices , 2008, 0812.2432.

[12]  Daniel M. Kane,et al.  Bounded Independence Fools Degree-2 Threshold Functions , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[13]  H. Yau,et al.  Bulk universality for generalized Wigner matrices , 2010, 1001.3453.

[14]  Sham M. Kakade,et al.  A tail inequality for quadratic forms of subgaussian random vectors , 2011, ArXiv.

[15]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[16]  T. Tao Topics in Random Matrix Theory , 2012 .

[17]  F. Barthe,et al.  Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems , 2012, 1202.5318.

[18]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[19]  R. Adamczak,et al.  Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order , 2013, 1304.1826.