Matrix Concentration: Order versus Anti-order

Abstract: The matrix Markov inequality by Ahlswede was stated using the Loewner anti-order between positive definite matrices. Wang use this to derive several other Chebyshev and Chernoff-type inequalities (Hoeffding, Bernstein, empirical Bernstein) in the Loewner anti-order, including self-normalized matrix martingale inequalities. These imply upper tail bounds on the maximum eigenvalue, such as those developed by Tropp and howard et al. The current paper develops analogs of all these inequalities in the Loewner order, rather than anti-order, by deriving a new matrix Markov inequality. These yield upper tail bounds on the minimum eigenvalue that are a factor of d tighter than the above bounds on the maximum eigenvalue.