# Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models

arXiv: https://arxiv.org/abs/2608.20183 · HTML: https://mlsanity.com/paper/2608.20183/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 62 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEs](https://mlsanity.com/paper/2101.08366/) (2021)
- [Pi-nets: Deep Polynomial Neural Networks](https://mlsanity.com/paper/2003.03828/) (2020)
- [Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition](https://mlsanity.com/paper/2310.06301/) (2023)
- [A Bayesian information criterion for singular models](https://mlsanity.com/paper/1309.0911/) (2013)
- [Polynomial Neural Fields for Subband Decomposition and Manipulation](https://mlsanity.com/paper/2302.04862/) (2023)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discovery](https://mlsanity.com/paper/2308.10283/) (2023)
- [From Global to Local: A Scalable Benchmark for Local Posterior Sampling](https://mlsanity.com/paper/2507.21449/) (2025)
- [Fast sampling and model selection for Bayesian mixture models](https://mlsanity.com/paper/2501.07668/) (2025)
- [Learning under Singularity: An Information Criterion improving WBIC and sBIC](https://mlsanity.com/paper/2402.12762/) (2024)

## Abstract (arXiv)

Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients λ, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.