How far are $d$-dimensional copulas with uniform $(d-1)$-marginals from (total) independence?
arXiv:2608.18286
2026
Regularization
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper proves that a joint distribution can have every (d-1)-dimensional marginal exactly independent while remaining substantially dependent in all d coordinates: the maximum uniform-copula deviation is 2^{-d}, and the maximum conditioning-based deviation is 1/3. The transferable asset is a sharp construction of dependence that is invisible to every lower-order marginal, providing controlled high-order-interaction data rather than merely pairwise-correlated data. This can be used both as a stress test for whether architectures actually learn d-way interactions and as an auxiliary representation loss that suppresses lower-order shortcuts while preserving a designated high-order signal.
Ideas from this paper
Unverified
2026
Construct training and evaluation examples whose every (d-1)-variable marginal is exactly independent, but whose full d-variable distribution contains a parity interaction. This isolates genuine high-order reasoning from shortcuts based on pairwise or lower-order statistics and can expose whether attention or MLP architectures learn the intended interaction.
Useful6/10
Difficulty3/10
Novelty6/10
Unverified
2026
Add an auxiliary loss that makes selected representation coordinates insensitive to all subsets of fewer than d variables while retaining a d-way parity statistic. The objective discourages the network from solving a task through pairwise shortcuts and explicitly rewards a controlled high-order interaction.
Useful5/10
Difficulty5/10
Novelty7/10