Core Concepts
This paper proposes a new randomized algorithm, called ContHutch++, that efficiently estimates the trace of any trace-class integral operator using operator-function products. The algorithm avoids discretization artifacts and provides rigorous high-probability error bounds.
Abstract
The paper introduces a continuous analogue of Hutchinson's estimator and Hutch++ for estimating the trace of a continuous, symmetric, positive semidefinite (PSD) function f(x,y) over a domain Ω.
Key highlights:
- Continuous Hutchinson's estimator: Uses Gaussian processes to approximate the trace of f, with a bias that decreases as the length-scale parameter ℓ→0.
- Continuous Hanson-Wright inequality: Provides a high-probability bound on the deviation of the continuous Hutchinson's estimator from its expected value.
- ContHutch++: Combines the continuous Hutchinson's estimator with a continuous randomized range finder to obtain an estimator that uses fewer operator-function products than continuous Hutchinson's estimator while maintaining rigorous error bounds.
- Applications: The authors demonstrate how ContHutch++ can be used to efficiently compute quantum density-of-states and estimate electromagnetic fields induced by incoherent sources.
The paper provides a comprehensive theoretical analysis of the proposed estimators, including explicit relationships between the accuracy, the number of operator-function products, and the intrinsic properties of the kernel f(x,y) such as regularity.