Recovering a separating set by thresholding the optimized DAT parameters

Prove that, under the generic regime described above, the solution \psi^* to the DAT separating representation search yields a true separating set for X and Y via thresholding: for any threshold c with 1 - \epsilon > c > \epsilon, the subset {Z_m : \psi^*_m > c} is a separating set SepSet(X, Y).

Background

DAT optimizes the soft-subset parameters \psi to drive a conditional-independence objective toward zero. Theorem 1 guarantees exact recovery of a separating set when variables with \psi_m=1 are taken, assuming thick-tailed noise distributions.

The conjecture strengthens this by proposing that, for generic noise choices, the optimized \psi* can be thresholded at any c in (\epsilon, 1-\epsilon) to retrieve the true separating set. Establishing this would justify practical thresholding rules and connect continuous \psi* solutions to discrete separating sets without relying on the specific thick-tail assumption.

References

In this case we conjecture ${Z_m}_{\psi_m*>c}$ is a separating set for any value of $1-\epsilon > c > \epsilon$.

— Scalable and Flexible Causal Discovery with an Efficient Test for Adjacency  (2406.09177 - Amin et al., 2024) in Appendix, Discussion of Theorem 1 (Theorem~\ref{Thm: main reliability})