Determine whether training-time control of disentanglement is possible

Determine whether the Intersection Euler Characteristic Profile can be used to control or steer neural-network training toward certified disentanglement.

Background

The paper’s differentiable-surrogate experiments fail to steer training toward lower certified entanglement: the tested surrogates are optimized while the exact Intersection ECP quotient increases. This leaves unresolved whether any training-time optimization method based on the interaction profile can reliably control certified disentanglement.

The issue is distinct from merely measuring disentanglement. The open question concerns using the topological statistic, or a faithful differentiable proxy for it, as an effective training objective.

References

$$ is integer-valued, and both differentiable surrogates we tried were optimized away from what the certificate measures (E6), so training-time control remains open.

— Certified Topological Interaction in Neural Representations: Class Disentanglement Is Mostly Pairwise  (2609.08561 - Majhi, 8 Sep 2026) in Section 6, subsection “What $\chi$ discards, and what $$ cannot steer”