Unknown eigenvalue structure for selecting approximation rank

Determine the number of large eigenvalues of the full-data kernel matrix used in the fast committor machine so as to inform the theoretically optimal choice of the RPCholesky approximation rank r, and develop practical methods to estimate this number from data.

Background

In the FCM, the RPCholesky method approximates the kernel matrix with rank r, trading off accuracy and computational cost. Theory suggests an optimal r depends on the count of large eigenvalues of the full kernel matrix.

The authors explicitly note that this eigenvalue information is not known in practice, leaving an unresolved question about how to determine or estimate it to guide rank selection.