The Lower Tail of Random Quadratic Forms, via the PAC Bayesian Method

Friday, April 17, 2015 - 9:00am - 9:50am
Keller 3-180
Roberto Oliveira (Institute of Pure and Applied Mathematics (IMPA))
Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the lower tail of such a matrix, and prove that it is subgaussian under a simple fourth moment assumption on the one-dimensional marginals of the random vectors. A similar result holds for more general sums of random positive semidefinite matrices and the (relatively simple) proof uses a variant of the so-called PAC-Bayesian method for bounding empirical processes. Some applications of the main result to high dimensional problems will be briefly sketched.
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