Copula Active Subspaces I: A Score-Covariance Method for Reduced-Order Non-Gaussian Density Estimation

In Bayesian inference problems with non-Gaussian observation noise, the posterior is only as accurate as the noise density, and gradient-based samplers need that density and its gradient evaluable pointwise, whether from an explicit expression or from code, and without an inner solve. We propose Copula Active Subspaces (CAS) to represent this noise density. A componentwise rank transform isolates the noise law's dependence in its copula, and a rank-r reduction keeps only the directions along which that dependence varies. These directions are the leading eigenvectors of the copula score covariance boldsymbolC := Cov_π_boldsymbolZ(nablalog cZ), which is what makes the reduction a copula active subspace. Because boldsymbolC vanishes when the coordinates are independent, these are directions of dependence, which the covariance of the data need not identify. From this construction follow a Gaussian-reference KL divergence bound with the explicit constant tfrac12, minimized over all rank-r reductions by exactly this eigenspace; a diagnostic for the error the reduction leaves behind, computable from the samples alone; and, from Hermite score matching, a reduced log-density and gradient in closed form, with the truncation orders and the Stage-2 regularization constants chosen on validation samples. The reduction replaces a d-dimensional density estimation problem by an r-dimensional one. On a d=20 noise law and a Bayesian inference problem with that noise, CAS lowers noise KL divergence more than fivefold and posterior KL divergence more than sevenfold against Gaussian-copula, product-of-marginals, and PCA-subspace baselines, and lowers noise KL divergence by factors of about 3.5 and 2.7 on two further d=20 examples.