Universality of Singular Complexity for Hyvärinen Generalized Bayes: Exact Transfer in Gaussian Factor Analysis

Manoj Saravanan, Rohit Kumar Salla
Proceedings of The 1st Symposium on Probabilistic Machine Learning, PMLR 327:50-85, 2026.

Abstract

Singular statistical models are asymptotically governed by local birational invariants such as the real log canonical threshold (RLCT), rather than by ambient parameter dimension. We study this singular complexity under generalized Bayes updates based on the Hyvärinen loss. We first prove a local comparison theorem: nonnegative analytic excess-loss germs that are locally comparable near a common zero set have the same local RLCT pair. As a corollary, losses that share an analytic minimizer map and have a nondegenerate quadratic germ lie in the same singular universality class. We then show that in analytic zero-mean Gaussian covariance models, both the population excess Gaussian log-loss and the excess Hyvärinen loss are locally equivalent to $\lVert \Sigma(\theta)- \Sigma_0 \rVert_{\mathrm{F}}^2$. Consequently, ordinary Gaussian Bayes and Hyvärinen generalized Bayes have identical local RLCT pairs at every covariance fiber. Applying this transfer principle to Gaussian factor analysis yields exact transfer of known local learning-coefficient results from ordinary Gaussian Bayes. In the one-factor model, the Hyvärinen posterior therefore has coefficients $p$, $(2p-1)/2$, and $3p/4$ on the corresponding covariance strata. Numerical experiments confirm the predicted local quadratic equivalence and the cancellation of the shared $ \lambda \log n$ term in paired free-energy differences.

Cite this Paper

BibTeX
@InProceedings{pmlr-v327-saravanan26a, title = {Universality of Singular Complexity for Hyvärinen Generalized Bayes: Exact Transfer in Gaussian Factor Analysis}, author = {Saravanan, Manoj and Salla, Rohit Kumar}, booktitle = {Proceedings of The 1st Symposium on Probabilistic Machine Learning}, pages = {50--85}, year = {2026}, editor = {Swaroop, Siddharth and Rügamer, David and Kristiadi, Agustinus}, volume = {327}, series = {Proceedings of Machine Learning Research}, month = {05 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v327/main/assets/saravanan26a/saravanan26a.pdf}, url = {https://proceedings.mlr.press/v327/saravanan26a.html}, abstract = { Singular statistical models are asymptotically governed by local birational invariants such as the real log canonical threshold (RLCT), rather than by ambient parameter dimension. We study this singular complexity under generalized Bayes updates based on the Hyvärinen loss. We first prove a local comparison theorem: nonnegative analytic excess-loss germs that are locally comparable near a common zero set have the same local RLCT pair. As a corollary, losses that share an analytic minimizer map and have a nondegenerate quadratic germ lie in the same singular universality class. We then show that in analytic zero-mean Gaussian covariance models, both the population excess Gaussian log-loss and the excess Hyvärinen loss are locally equivalent to $\lVert \Sigma(\theta)- \Sigma_0 \rVert_{\mathrm{F}}^2$. Consequently, ordinary Gaussian Bayes and Hyvärinen generalized Bayes have identical local RLCT pairs at every covariance fiber. Applying this transfer principle to Gaussian factor analysis yields exact transfer of known local learning-coefficient results from ordinary Gaussian Bayes. In the one-factor model, the Hyvärinen posterior therefore has coefficients $p$, $(2p-1)/2$, and $3p/4$ on the corresponding covariance strata. Numerical experiments confirm the predicted local quadratic equivalence and the cancellation of the shared $ \lambda \log n$ term in paired free-energy differences. } }
Endnote
%0 Conference Paper %T Universality of Singular Complexity for Hyvärinen Generalized Bayes: Exact Transfer in Gaussian Factor Analysis %A Manoj Saravanan %A Rohit Kumar Salla %B Proceedings of The 1st Symposium on Probabilistic Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Siddharth Swaroop %E David Rügamer %E Agustinus Kristiadi %F pmlr-v327-saravanan26a %I PMLR %P 50--85 %U https://proceedings.mlr.press/v327/saravanan26a.html %V 327 %X Singular statistical models are asymptotically governed by local birational invariants such as the real log canonical threshold (RLCT), rather than by ambient parameter dimension. We study this singular complexity under generalized Bayes updates based on the Hyvärinen loss. We first prove a local comparison theorem: nonnegative analytic excess-loss germs that are locally comparable near a common zero set have the same local RLCT pair. As a corollary, losses that share an analytic minimizer map and have a nondegenerate quadratic germ lie in the same singular universality class. We then show that in analytic zero-mean Gaussian covariance models, both the population excess Gaussian log-loss and the excess Hyvärinen loss are locally equivalent to $\lVert \Sigma(\theta)- \Sigma_0 \rVert_{\mathrm{F}}^2$. Consequently, ordinary Gaussian Bayes and Hyvärinen generalized Bayes have identical local RLCT pairs at every covariance fiber. Applying this transfer principle to Gaussian factor analysis yields exact transfer of known local learning-coefficient results from ordinary Gaussian Bayes. In the one-factor model, the Hyvärinen posterior therefore has coefficients $p$, $(2p-1)/2$, and $3p/4$ on the corresponding covariance strata. Numerical experiments confirm the predicted local quadratic equivalence and the cancellation of the shared $ \lambda \log n$ term in paired free-energy differences.
APA
Saravanan, M. & Salla, R.K.. (2026). Universality of Singular Complexity for Hyvärinen Generalized Bayes: Exact Transfer in Gaussian Factor Analysis. Proceedings of The 1st Symposium on Probabilistic Machine Learning, in Proceedings of Machine Learning Research 327:50-85 Available from https://proceedings.mlr.press/v327/saravanan26a.html.

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