What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory

TL;DR

This paper introduces a variational framework and a new function space to analyze the properties of functions learned by deep neural networks with ReLU activations, linking neural networks to spline theory and sparsity.

Contribution

It develops a novel variational approach and function space that characterize deep neural network solutions, connecting architectural features to function regularization and spline theory.

Findings

Deep ReLU networks are solutions to regularized data fitting in the proposed function space.

The function space captures compositional structure and sparsity, explaining architectural choices.

The framework links neural networks to classical spline theory, providing new theoretical insights.

Abstract

We develop a variational framework to understand the properties of functions learned by fitting deep neural networks with rectified linear unit activations to data. We propose a new function space, which is reminiscent of classical bounded variation-type spaces, that captures the compositional structure associated with deep neural networks. We derive a representer theorem showing that deep ReLU networks are solutions to regularized data fitting problems over functions from this space. The function space consists of compositions of functions from the Banach spaces of second-order bounded variation in the Radon domain. These are Banach spaces with sparsity-promoting norms, giving insight into the role of sparsity in deep neural networks. The neural network solutions have skip connections and rank bounded weight matrices, providing new theoretical support for these common architectural choices. The variational problem we study can be recast as a finite-dimensional neural network training problem with regularization schemes related to the notions of weight decay and path-norm regularization. Finally, our analysis builds on techniques from variational spline theory, providing new connections between deep neural networks and splines.