arXiv:2607.17630 2026 Regularization 2 ideas extracted · analyzed Aug 30, 2026
The paper develops a constructive hierarchy of finite Hankel-matrix obstructions derived from conditional positive definiteness of free cumulants. The transferable asset is the use of low-order determinant or eigenvalue certificates as inexpensive structural constraints on a learned sequence of moments or cumulants. These constraints can regularize neural distribution heads, latent-variable models, or learned samplers by penalizing outputs that cannot correspond to a structurally valid distribution. The tests are necessary rather than sufficient, so they should be used as validity regularizers and diagnostics, not as complete proofs of free infinite divisibility.
Attach finite Hankel positive-semidefiniteness penalties to a neural model that predicts scalar moments, cumulants, or beta-distribution parameters. The exact beta inequality supplies a very cheap first-stage barrier, while eigenvalue penalties on larger Hankel matrices constrain higher-order structure.
Exploit the paper's nested obstruction hierarchy by applying cheap low-order Hankel tests to every example and evaluating larger matrices only for outputs near the current feasibility boundary. This turns higher-order structural validation into an adaptive curriculum rather than an always-on expensive eigendecomposition.