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Theory of linear rescaling and nonlinear extensions

Characterize and mathematically describe why the linear rescaling via the scaling matrix M in the fast committor machine is so successful, and investigate possible nonlinear extensions of this rescaling to further improve performance.

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Background

A central component of FCM is an adaptive linear transformation (scaling matrix M) informed by gradients of the target function. While effective empirically, the authors explicitly identify understanding its success and potential nonlinear generalizations as a deep, unresolved question.

Addressing this would provide theoretical foundations for the feature transformation and could lead to enhanced versions using nonlinear mappings.

References

From a mathematical perspective, there remain several challenging questions about the FCM's performance. Last, there is the deep question about why the linear rescaling is so successful, how to describe it mathematically, and what nonlinear extensions would be possible.

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