Fisher Completions and Statistical Regularization of Wrapped Gaussian Families
Abstract
We determine the intrinsic Fisher completion of the full-covariance wrapped Gaussian family on a fixed torus and relate its geometry to uniform estimation. The completion is homeomorphic to the total-variation closure. Every added law is, in suitable integer torus coordinates, a product of a uniform law and a lower-dimensional wrapped Gaussian. All these densities are positive and real analytic, but in dimension at least two no added point has a compact neighborhood. Small balls remain contractible, and explicit minimizing geodesics pass through added points. In dimension at least two, every fixed Fisher neighborhood of an added point has a positive minimax risk at every sample size. For subfamilies with a common positive lower bound on the variance in every nonzero integer direction, we prove that compactness of the completed closure is equivalent to uniform consistency in either L1 or completed-Fisher loss. Equivalently, the Fourier moments must decay uniformly over the subfamily. A fixed covariance floor guarantees this condition without an upper covariance bound and gives L1 risk at most Cε,dn−1/2{log(n+1)}d/4. A common-phase example illustrates how partially uniform limits can retain relative-phase information while remaining uniformly estimable.
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BibTeX
@misc{cho2026fisher,
title = {{Fisher Completions and Statistical Regularization of Wrapped Gaussian Families}},
author = {Gunhee Cho and Hyun Chul Jang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37526},
eprint = {2610.37526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37526},
}
RIS
TY - UNPB
TI - Fisher Completions and Statistical Regularization of Wrapped Gaussian Families
AU - Gunhee Cho
AU - Hyun Chul Jang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37526
UR - https://arxiv.org/abs/2610.37526
AB - We determine the intrinsic Fisher completion of the full-covariance wrapped Gaussian family on a fixed torus and relate its geometry to uniform estimation. The completion is homeomorphic to the total-variation closure. Every added law is, in suitable integer torus coordinates, a product of a uniform law and a lower-dimensional wrapped Gaussian. All these densities are positive and real analytic, but in dimension at least two no added point has a compact neighborhood. Small balls remain contractible, and explicit minimizing geodesics pass through added points. In dimension at least two, every fixed Fisher neighborhood of an added point has a positive minimax risk at every sample size. For subfamilies with a common positive lower bound on the variance in every nonzero integer direction, we prove that compactness of the completed closure is equivalent to uniform consistency in either L^1 or completed-Fisher loss. Equivalently, the Fourier moments must decay uniformly over the subfamily. A fixed covariance floor guarantees this condition without an upper covariance bound and gives L^1 risk at most C_ε,dn^−1/2{log(n+1)}^d/4. A common-phase example illustrates how partially uniform limits can retain relative-phase information while remaining uniformly estimable.
ER -