Lower bounds for trace estimation via Block Krylov and other methods

TL;DR

This paper establishes theoretical lower bounds for trace estimation of matrix functions, analyzing the number of Krylov steps needed and connecting it to polynomial approximation limits, especially for inverse and Wishart matrices.

Contribution

It introduces fundamental lower bounds on the number of queries and Krylov steps required for trace estimation, linking these to polynomial approximation theory.

Findings

Derived upper bounds on Krylov steps for specific matrix functions.

Established lower bounds on the number of queries for trace estimation.

Clarified the relationship between Krylov methods and polynomial approximation degree.

Abstract

This paper studies theoretical lower bounds for estimating the trace of a matrix function, $\text{tr}(f(A))$, focusing on methods that use Hutchinson's method along with Block Krylov techniques. These methods work by approximating matrix-vector products like $f(A)V$ using a Block Krylov subspace. This is closely related to approximating functions with polynomials. We derive theoretical upper bounds on how many Krylov steps are needed for functions such as $A^{-1/2}$ and $A^{-1}$ by analyzing the upper bounds from the polynomial approximation of their scalar equivalent. In addition, we also develop lower limits on the number of queries needed for trace estimation, specifically for $\text{tr}(W^{-p})$ where $W$ is a Wishart matrix. Our study clarifies the connection between the number of steps in Block Krylov methods and the degree of the polynomial used for approximation. This links the total cost of trace estimation to basic limits in polynomial approximation and how much information is needed for the computation.