Deep Sturm–Liouville: From Sample-Based to 1D Regularization with Learnable Orthogonal Basis Functions

Although Artificial Neural Networks (ANNs) have achieved remarkable success across various tasks, they still suffer from limited generalization. We hypothesize that this limitation arises from the traditional sample-based (0--dimensionnal) regularization used in ANNs. To overcome this, we introduce Deep Sturm-Liouville (DSL), a novel function approximator that enables continuous 1D regularization along field lines in the input space by integrating the Sturm-Liouville Theorem (SLT) into the deep learning framework. DSL defines field lines traversing the input space, along which a Sturm-Liouville problem is solved to generate orthogonal basis functions, enforcing implicit regularization thanks to the desirable properties of SLT. These basis functions are linearly combined to construct the DSL approximator. Both the vector field and basis functions are parameterized by neural networks and learned jointly. We demonstrate that the DSL formulation naturally arises when solving a Rank-1 Parabolic Eigenvalue Problem. DSL is trained efficiently using stochastic gradient descent via implicit differentiation and achieves competitive performance on diverse multivariate datasets, including high-dimensional image datasets such as MNIST and CIFAR-10.

In traditional machine learning, data is typically projected onto a predefined target representation with the goal of learning useful features. In contrast, Deep Sturm—Liouville introduces a fundamentally different perspective: inspired by fluid dynamics, we define the transformation directly within the data space itself. This novel perspective enables the design of novel regularization mechanisms. As a result, we achieve improved sample efficiency—a key indicator of better generalization.