Point-to-Manifold Geometry: Flexibly Overcoming the Curse of Dimensionality in Neural Computational Units

The ability of neural networks to generalize is fundamentally shaped by their computational units. While the perceptron provides a first-order, linear inductive bias and the radial basis function (RBF) unit offers an isotropic curved bias, we identify a critical opportunity for a structured extension: the Generative Matching Unit (GMU). Each individual GMU captures complex dependencies by treating the forward pass as an inference problem; its internal generative model optimizes instance-specific latent parameters to compute a reconstruction error, effectively measuring a point-to-manifold distance as opposed to the point-to-point distance in RBFs. We focus on linear GMUs, which yield closed-form analytical expressions for fast computation and extend naturally to convolutional architectures. Our theoretical analysis demonstrates that linear GMUs are highly flexible universal approximators capable of recovering structured class posteriors. We prove that while a single GMU can exactly emulate an RBF kernel and two can emulate the decision boundary of a perceptron, emulating a single $k$-order GMU’s decision boundary requires a hidden layer of standard units that scales polynomially or exponentially with input dimensionality. Furthermore, we prove that the GMU’s point-to-manifold distance remains discriminative in high dimensions where standard Euclidean distances fail, offering unique robustness to the curse of dimensionality. Finally, we show that GMUs offer superior flexibility and efficiency in learning smooth manifold decision boundaries compared to MLPs and RBFs. Motivated by the theoretical results, we place GMUs in the first network layer like RBFs, where their role is to enhance linear separability for subsequent layers. This avoids the typical gradient collapse problem with stacking RBF-like units while ensuring the generalization benefits remain. We find that across extensive experiments involving 27 tabular datasets, five vision datasets evaluated across 34 test-time corruption settings, and 30 synthetic scenarios, GMU networks demonstrate statistically significant improvements in generalization and robustness.