Singular Bayesian Neural Networks

Bayesian neural networks promise calibrated uncertainty but require $O(mn)$ parameters for standard mean-field Gaussian posteriors. We argue this cost is often unnecessary, particularly when weight matrices exhibit fast singular value decay. By parameterizing weights as $W = AB^{\top}$ with $A \in \mathbb{R}^{m \times r}$, $B \in \mathbb{R}^{n \times r}$, we induce a posterior that is singular with respect to the Lebesgue measure, concentrating on the rank-$r$ manifold. This singularity captures structured weight correlations through shared latent factors, geometrically distinct from mean-field's independence assumption. We derive PAC-Bayes generalization bounds whose complexity term scales as $\sqrt{r(m+n)}$ instead of $\sqrt{m n}$, and prove loss bounds that decompose the error into optimization and rank-induced bias using the Eckart-Young-Mirsky theorem. We further adapt recent Gaussian complexity bounds for low-rank deterministic networks to Bayesian predictive means by exploiting their stability under convex combinations. Empirically, across MLPs, LSTMs, and Transformers on standard benchmarks, our method achieves predictive performance competitive with 5-member Deep Ensembles while using up to $15\times$ fewer parameters. Furthermore, it substantially improves OOD detection and often improves calibration relative to mean-field and perturbation baselines.