Transversality and Geometric Regularisation in Distributional Statistical Models
arXiv:2605.04536
The paper develops a statistical framework that replaces classical densities with tempered distribution‑kernel pairs, showing that the smoothing kernel acts as a geometric regulariser that puts parametric models in generic transversal position relative to degeneracy loci, and provides rank‑based criteria to verify this for several model families.
Abstract
The distributional statistical framework replaces classical probability densities by distribution-kernel pairs , where is a tempered distribution and 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.
29 pages, no figures no tables. In the second version some sketches were replaced by proofs, an example of M-determinancy was added. In the third version the model representation becomes a tempered distribution instead of a pari tempered distribution and a Swartz kernel. In the fourth version the example oinvolving the Cauchy distribution was corrected