The authors analyze Bayesian Kolmogorov-Arnold Networks (KANs)—architectures that replace standard neural network edges with learnable spline functions—and prove posterior contraction rates (how fast the Bayesian posterior concentrates around the true function) over anisotropic Besov spaces (function spaces where smoothness can vary across dimensions). They show that sparse Bayesian KANs with spike-and-slab priors achieve near-minimax rates, and that by placing a hyperprior on a single model-size parameter, the posterior adapts to unknown smoothness and still gets the optimal rate. Unlike standard MLPs, KANs can keep depth fixed and control complexity via width, spline resolution, and parameter sparsity.
Main takeaways:
- KANs use learnable spline functions on edges instead of fixed activation functions; this paper gives the first rigorous Bayesian statistical analysis of them.
- Sparse Bayesian KANs (with spike-and-slab priors on parameters) achieve near-minimax posterior contraction rates over anisotropic Besov spaces (functions with dimension-dependent smoothness).
- A hyperprior on a single model-size parameter lets the posterior adapt to unknown smoothness without sacrificing the rate.
- Unlike MLPs, KANs can keep depth fixed and control approximation complexity via width, spline-grid range, and sparsity.
- The results extend to compositional Besov spaces (functions with layered structure), where rates depend on layer-wise smoothness and effective dimension, avoiding the curse of dimensionality.