Transversality and Geometric Regularisation in Distributional Statistical Models

TL;DR

This paper introduces a geometric regularisation approach using transversality theorems to improve the robustness and identifiability of distributional statistical models, with applications to various parametric models.

Contribution

It develops a finite-dimensional transversality theorem for generic kernels, providing verifiable conditions to ensure models avoid degeneracy, enhancing understanding of model regularity and stability.

Findings

Proves a weak transversality theorem for generic kernels in rich families.

Provides Jacobian rank conditions to verify transversality hypotheses.

Applies results to location families, log-normal, Stein discrepancies, and graphical models.

Abstract

The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, \varphi)$, where $T$ is a tempered distribution and $\varphi$ is a rapidly decaying kernel. We develop the thesis that the kernel acts as a geometric regulariser, placing parametric statistical models in generic (transversal) position relative to degeneracy loci encoding non-identifiability, singular information, moment indeterminacy, and representation failure. Using the transversality theorems of Whitney, Thom, and Mather, we prove a finite-dimensional weak transversality theorem: for a generic kernel in any sufficiently rich family, the kernel-induced feature map avoids degeneracy strata of sufficiently high codimension. We establish verifiable conditions -- formulated as rank conditions on the Jacobian of the joint feature map -- under which the transversality hypothesis can be checked, and verify them for location families, the log-normal, Stein discrepancies, and graphical models. The present results apply to parametric models; extensions to semiparametric and nonparametric settings are discussed. The degeneracy classification includes representation degeneracy (Type 0) for models without closed-form densities and higher-order instabilities (Type IV) in non-chordal graphical models. Identifiability, robustness, moment determinacy, Fisher information regularity, Stein discrepancy, inferential separation, and the Behrens-Fisher problem all admit a unified geometric interpretation as transversality conditions on the feature map. This paper serves as a geometric companion to a series of papers developing the distributional framework.