Matrix Concentration: Order versus Anti-order

TL;DR

This paper develops new matrix concentration inequalities in the Loewner order, providing tighter bounds on the minimum eigenvalue compared to existing bounds on the maximum eigenvalue, by extending the matrix Markov inequality.

Contribution

It introduces a novel matrix Markov inequality in the Loewner order, enabling the derivation of eigenvalue bounds that are tighter than previous results based on the anti-order.

Findings

Tighter upper tail bounds on the minimum eigenvalue by a factor of d.

Extension of Chebyshev and Chernoff-type inequalities to the Loewner order.

Development of analogs for existing inequalities in the Loewner 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.