474 research outputs found
Concentration inequalities for random tensors
We show how to extend several basic concentration inequalities for simple
random tensors where all are
independent random vectors in with independent coefficients. The
new results have optimal dependence on the dimension and the degree . As
an application, we show that random tensors are well conditioned: independent copies of the simple random tensor
are far from being linearly dependent with high probability. We prove this fact
for any degree and conjecture that it is true for any
.Comment: A few more typos were correcte
A note on the Hanson-Wright inequality for random vectors with dependencies
We prove that quadratic forms in isotropic random vectors in
, possessing the convex concentration property with constant ,
satisfy the Hanson-Wright inequality with constant , where 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-Gin\'e and Ledoux-Talagrand). Previous results
of this type relied on stronger isoperimetric properties of 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
-valued Gaussian variables due to Koltchinskii and Lounici
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