Correct the scale sensitivity of covariance matching

Determine whether scale-normalizing the covariance-matching objective or retuning its fit-loss weight separately for graph size and edge density removes the whole-graph recovery degradation without destroying the directional separation guaranteed by covariance matching.

Background

Covariance matching improves directional identifiability but produces substantially worse whole-graph structural Hamming distance in many experimental settings. The paper attributes part of this cost to the growth of raw covariance scale with graph size and density while the fit-loss weight remains fixed.

The authors did not evaluate a scale-normalized covariance objective or retune the fit-loss weight separately for each graph-size and density condition. The unresolved issue is whether either intervention can remove the whole-graph cost while preserving the strict directional separation established for covariance matching.

References

Section~\ref{sec:emp-tradeoff} traces covariance matching's worse SHD in part to raw-scale growth with graph density interacting with a fixed loss weight $w_f$, but we have not implemented or evaluated a scale-normalized variant of the covariance objective (for instance, dividing $L$ by $\mathrm{tr}(\hat\Sigma)$ or $|\hat\Sigma|_F2$) or an ablation retuning $w_f$ separately per $(d,\text{density})$ cell; doing either, and confirming it removes the effect without disturbing Lemma~\ref{lem:separation}'s separation, is left to future work.

— Guide, Not Bind: Why Defeasible Priors Fail in Augmented Lagrangian Causal Discovery  (2609.03442 - Sundararaman et al., 3 Sep 2026) in Limitations, item 4; Section 6.4