Yes, the ideas presented in this paper can indeed be applied to estimate other functionals of integral operators, such as the log-determinant or eigenvalue counts, in a discretization-oblivious manner. Here’s how:
Generalization of the Estimation Framework: The continuous trace estimation framework established by ContHutch++ can be generalized to estimate various functionals of integral operators. For instance, the log-determinant of an operator can be expressed in terms of its eigenvalues, and since the trace is a linear functional, similar stochastic techniques can be employed to estimate the log-determinant by leveraging the relationship between the trace and the eigenvalues.
Use of Operator-Function Products: The operator-function product approach used in the continuous trace estimation can be adapted to compute other functionals. For example, to estimate the log-determinant, one could utilize the identity that relates the log-determinant to the trace of the logarithm of the operator, i.e., (\\log \\det(F) = \ ext{tr}(\\log(F))). This allows for the application of the same stochastic sampling techniques to estimate the trace of the logarithm of the operator.
Eigenvalue Counting: For estimating eigenvalue counts, one can employ techniques similar to those used in the continuous trace estimation. By constructing appropriate probe functions and utilizing the properties of the integral operator, one can develop estimators that count the number of eigenvalues below a certain threshold, again without relying on specific discretization schemes.
Error Bounds and High-Probability Guarantees: The rigorous error bounds derived for the continuous trace estimation can be extended to these new functionals. By establishing connections between the properties of the integral operator and the desired functional, one can derive high-probability guarantees for the estimates, ensuring that they remain accurate even in the presence of discretization errors.
Applications in Quantum Mechanics and Statistical Physics: The ability to estimate functionals like the log-determinant and eigenvalue counts has significant implications in fields such as quantum mechanics and statistical physics, where these quantities often play critical roles in understanding system behavior. The proposed methods can thus facilitate more efficient computations in these domains.
In summary, the methodologies developed in this paper provide a versatile framework that can be adapted to estimate a variety of functionals of integral operators, enhancing their applicability across different scientific and engineering disciplines while maintaining robustness against discretization errors.