arXiv:2608.01516 2026 Regularization 1 ideas extracted · analyzed Aug 31, 2026
The paper provides a dimension-free anti-concentration principle for vector-valued polynomial pushforwards: an absolutely continuous polynomial image of a log-concave distribution cannot place too much mass in a small output set. Its most transferable construction is a nondegeneracy certificate based on the covariance of monomials in normalized output coordinates, rather than only raw covariance or Jacobian rank. This suggests a representation regularizer that prevents hidden features from collapsing in nonlinear directions that ordinary whitening misses. The theorem motivates the target, while a regularized log-determinant of the empirical polynomial-feature covariance provides a practical differentiable surrogate.
Apply a low-degree polynomial feature lift to normalized hidden representations and penalize degeneracy of the covariance in that lifted space. This can detect collapse in nonlinear combinations of features even when the raw hidden covariance appears healthy.