Transversality and Geometric Regularisation in Distributional Statistical Models
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910253237403648 |
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| author | Labouriau, R. |
| author_facet | Labouriau, R. |
| contents | The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, φ)$, where $T$ is a tempered distribution and $φ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04536 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transversality and Geometric Regularisation in Distributional Statistical Models Labouriau, R. Statistics Theory Differential Geometry Methodology 62B05, 62F35, 57R45, 46F05, 53B12 The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, φ)$, where $T$ is a tempered distribution and $φ$ 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. |
| title | Transversality and Geometric Regularisation in Distributional Statistical Models |
| topic | Statistics Theory Differential Geometry Methodology 62B05, 62F35, 57R45, 46F05, 53B12 |
| url | https://arxiv.org/abs/2605.04536 |