🤖 AI Summary
This paper investigates theoretical lower bounds for trace estimation of matrix functions, focusing on query complexity and computational cost of the Hutchinson method and Block Krylov subspace techniques. By establishing a tight theoretical connection between the number of Block Krylov iterations and the degree of polynomial approximation to scalar functions, the work derives the first query-complexity lower bound for estimating (mathrm{tr}(W^{-p})) under the Wishart matrix model. Concurrently, it obtains matching upper bounds on the required Krylov iteration count for key matrix functions including (A^{-1/2}) and (A^{-1}). The results uncover a fundamental trade-off among polynomial approximation accuracy, number of random queries, and Krylov iteration cost. This provides the first rigorous complexity characterization of trace estimation algorithms grounded in random matrix theory, thereby filling a long-standing gap in the theoretical understanding of lower bounds for stochastic trace estimators.
📝 Abstract
This paper studies theoretical lower bounds for estimating the trace of a matrix function, $ ext{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 $ ext{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.