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

@article{Parhi2022WhatKO,
title={What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory},
author={Rahul Parhi and Robert D. Nowak},
journal={ArXiv},
year={2022},
volume={abs/2105.03361}
}
• Published 7 May 2021
• Computer Science
• ArXiv
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…
12 Citations

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## References

SHOWING 1-10 OF 61 REFERENCES
Deep Neural Networks With Trainable Activations and Controlled Lipschitz Constant
• Computer Science
IEEE Transactions on Signal Processing
• 2020
It is proved that there always exists a solution that has continuous and piecewise-linear (linear-spline) activations and an $\ell _1$ penalty on the parameters of the activations favors the learning of sparse nonlinearities.
Learning Activation Functions in Deep (Spline) Neural Networks
• Computer Science
IEEE Open Journal of Signal Processing
• 2020
An efficient computational solution to train deep neural networks with free-form activation functions by using an equivalent B-spline basis to encode the activation functions and by expressing the regularization as an $\ell _1$-penalty.
A representer theorem for deep neural networks
• M. Unser
• Computer Science
J. Mach. Learn. Res.
• 2019
A general representer theorem for deep neural networks is derived that makes a direct connection with splines and sparsity, and it is shown that the optimal network configuration can be achieved with activation functions that are nonuniform linear splines with adaptive knots.
A Unifying Representer Theorem for Inverse Problems and Machine Learning
• M. Unser
• Mathematics
Found. Comput. Math.
• 2021
A general representer theorem is presented that characterizes the solutions of a remarkably broad class of optimization problems and is used to retrieve a number of known results in the literature---e.g., the celebrated representser theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes.
Are wider nets better given the same number of parameters?
• Computer Science
ICLR
• 2021
It is shown that for models initialized with a random, static sparsity pattern in the weight tensors, network width is the determining factor for good performance, while the number of weights is secondary, as long as trainability is ensured.
Banach Space Representer Theorems for Neural Networks and Ridge Splines
• Computer Science, Mathematics
J. Mach. Learn. Res.
• 2021
A variational framework to understand the properties of the functions learned by neural networks fit to data and a representer theorem showing that finite-width, single-hidden layer neural networks are solutions to inverse problems with total variation-like regularization is derived.
Mad Max: Affine Spline Insights Into Deep Learning
• Computer Science
Proceedings of the IEEE
• 2021
A rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators is built and a simple penalty term is proposed that can be added to the cost function of any DN learning algorithm to force the templates to be orthogonal with each other.
Pufferfish: Communication-efficient Models At No Extra Cost
• Computer Science
ArXiv
• 2021
PUFFERFISH is a communication and computation efficient distributed training framework that incorporates the gradient compression into the model training process via training low-rank, pre-factorized deep networks and leads to equally accurate, small-parameter models while avoiding the burden of “winning the lottery”.
Pufferfish: Communication-efficient models at no 26 R
• PARHI AND R. D. NOWAK extra cost, Proceedings of Machine Learning and Systems, 3
• 2021