Preference for exponential kernel over squared exponential

Determine the mathematical or statistical reasons that the exponential kernel k_M(x,x') used in the fast committor machine yields superior performance compared to the squared exponential kernel and other kernel choices for committor approximation.

Background

FCM uses an exponential kernel and reports empirical advantages over the squared exponential kernel. The authors explicitly list understanding this preference as a challenging open question.

Clarifying the conditions under which the exponential kernel is preferable would guide kernel selection and potentially lead to principled kernel design for committor problems.

References

From a mathematical perspective, there remain several challenging questions about the FCM's performance. These include understanding why the method takes so few iterations to successfully converge and what makes the exponential kernel e:kMkernel preferable over the square exponential kernel and other choices.

— The fast committor machine: Interpretable prediction with kernels (2405.10410 - Aristoff et al., 16 May 2024) in Section 4 (Conclusion)