Develop a differentiable surrogate that tracks certified interaction

Develop a differentiable surrogate for the Intersection Euler Characteristic Profile interaction quotient that provably tracks certified class disentanglement during neural-network optimization.

Background

The paper tests two natural differentiable surrogates for the interaction quotient. One can be reduced by inflating feature norms, while the other can be reduced by increasing within-class dispersion; both therefore move in the opposite direction from the certified Intersection ECP entanglement.

The experiments show that optimizing a smooth proxy does not currently provide a reliable way to steer training toward topologically certified disentanglement. A successful surrogate would need to avoid both scale and dispersion exploits while maintaining a provable relationship to the non-differentiable certified statistic.