A Lyapunov type theorem from Kadison-Singer

TL;DR

This paper extends the Kadison-Singer theorem by demonstrating that any convex combination of certain matrices can be closely approximated by a subset sum, improving the approximation bounds in operator norm.

Contribution

It generalizes the Kadison-Singer result to approximate any convex combination of matrices with subset sums, achieving a better approximation order.

Findings

Approximation in operator norm to order  with subset sums.

Extension of Kadison-Singer to convex combinations.

Improved bounds on matrix approximations.

Abstract

Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem by showing that if u_1, ..., u_m are column vectors in C^d such that \sum u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so that \sum_{i \in S} u_iu_i^* is approximately (1/2)I, with the approximation good in operator norm to order \epsilon^{1/2} where \epsilon = \max \|u_i\|^2. We extend their result to show that every linear combination of the matrices u_iu_i^* with coefficients in [0,1] can be approximated in operator norm to order \epsilon^{1/8} by a matrix of the form \sum_{i \in S} u_iu_i^*.