The ContHutch++ estimator, as presented, is specifically designed for continuous, symmetric positive semidefinite (PSD) functions. To extend this estimator to handle non-symmetric or non-PSD integral operators, several modifications would be necessary.
Generalization of the Kernel: The first step would involve generalizing the covariance kernel used in the Gaussian process. For non-symmetric functions, one could consider using a more general class of kernels that do not necessarily enforce symmetry. This would allow the estimator to accommodate the asymmetry in the operator.
Modification of the Randomized Range Finder: The randomized range finder, which is a crucial component of the ContHutch++ algorithm, would need to be adapted to account for the lack of symmetry. This could involve using a different approach to construct the orthonormal basis that captures the essential features of the non-symmetric operator.
Error Analysis: The error bounds derived for the current version of ContHutch++ rely on properties specific to symmetric PSD functions. For non-symmetric or non-PSD operators, new error bounds would need to be established. This would likely involve a more complex analysis that takes into account the spectral properties of the non-symmetric operator.
Application of Non-Symmetric Techniques: Techniques from the theory of non-symmetric matrices, such as the use of singular value decomposition (SVD) or generalized eigenvalue problems, could be integrated into the estimator. This would allow for the effective handling of non-symmetric operators while still leveraging the benefits of the randomized approach.
By implementing these modifications, the ContHutch++ estimator could be adapted to estimate traces of a broader class of integral operators, thus enhancing its applicability in various fields, including quantum mechanics and numerical analysis.