Tail Bounds for All Eigenvalues of a Sum of Random Matrices.

摘要

The field of nonasymptotic random matrix theory has traditionally focused on the problem of bounding the extreme eigenvalues of a random matrix. In some circumstances, however, we may also be interested in studying the behavior of the interior eigenvalues. In this case, classical tools do not readily apply. Indeed, the interior eigenvalues are determined by the minmax of a random process, which is very challenging to control. This paper demonstrates that it is possible to combine the matrix Laplace transform method detailed in Tro11c with the Courant-Fischer characterization of eigenvalues to obtain nontrivial bounds on the interior eigenvalues of a sum of random self-adjoint matrices. This approach expands the scope of the matrix probability inequalities from Tro11c so that they provide interesting information about the bulk spectrum.