# Tubular Riemannian Laplace Approximations for Bayesian Neural Networks

arXiv: https://arxiv.org/abs/2512.24381 · HTML: https://mlsanity.com/paper/2512.24381/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 30 Dec 2025.

## Builds on (most cited first, 20 of 20)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Limitations of the Empirical Fisher Approximation for Natural Gradient Descent](https://mlsanity.com/paper/1905.12558/) (2019)
- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (2019)
- [Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning](https://mlsanity.com/paper/2104.04975/) (2021)
- [Adapting the Linearised Laplace Model Evidence for Modern Deep Learning](https://mlsanity.com/paper/2206.08900/) (2022)
- [Reparameterization invariance in approximate Bayesian inference](https://mlsanity.com/paper/2406.03334/) (2024)
- [The Geometry of Neural Nets' Parameter Spaces Under Reparametrization](https://mlsanity.com/paper/2302.07384/) (2023)
- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning](https://mlsanity.com/paper/2407.13711/) (2024)
- [Riemannian Laplace Approximation with the Fisher Metric](https://mlsanity.com/paper/2311.02766/) (2023)
- [Paths and Ambient Spaces in Neural Loss Landscapes](https://mlsanity.com/paper/2503.03382/) (2025)
- [Geometric Gaussian Approximations of Probability Distributions](https://mlsanity.com/paper/2507.00616/) (2025)

## Built on by (most cited first, 1 of 1)

- [Contrastive Linearized Laplace: Loss Geometry, Valid Repairs, and Reliable Cores](https://mlsanity.com/paper/2610.14875/) (2026)

## Abstract (arXiv)

Laplace approximations are among the simplest and most practical methods for approximate Bayesian inference in neural networks, yet their Euclidean formulation struggles with the highly anisotropic, curved loss surfaces and large symmetry groups that characterize modern deep models. Recent work has proposed Riemannian and geometric Gaussian approximations to adapt to this structure. Building on these ideas, we introduce the Tubular Riemannian Laplace (TRL) approximation. TRL explicitly models the posterior as a probabilistic tube that follows a low-loss valley induced by functional symmetries, using a Fisher/Gauss-Newton metric to separate prior-dominated tangential uncertainty from data-dominated transverse uncertainty. We interpret TRL as a scalable reparametrised Gaussian approximation that utilizes implicit curvature estimates to operate in high-dimensional parameter spaces. Our empirical evaluation on ResNet-18 (CIFAR-10 and CIFAR-100) demonstrates that TRL achieves excellent calibration, matching or exceeding the reliability of Deep Ensembles (in terms of ECE) while requiring only a fraction (1/5) of the training cost. TRL effectively bridges the gap between single-model efficiency and ensemble-grade reliability.