Approximation of Lipschitz Functions using Deep Spline Neural Networks

• URL: http://arxiv.org/abs/2204.06233v1
• Date: Wed, 13 Apr 2022 08:07:28 GMT
• Title: Approximation of Lipschitz Functions using Deep Spline Neural Networks
• Authors: Sebastian Neumayer and Alexis Goujon and Pakshal Bohra and Michael Unser
• Abstract summary: We propose to use learnable spline activation functions with at least 3 linear regions instead of ReLU networks. We prove that this choice is optimal among all component-wise $1$-Lipschitz activation functions. This choice is at least as expressive as the recently introduced non component-wise Groupsort activation function for spectral-norm-constrained weights.
• Score: 21.13606355641886
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Lipschitz-constrained neural networks have many applications in machine learning. Since designing and training expressive Lipschitz-constrained networks is very challenging, there is a need for improved methods and a better theoretical understanding. Unfortunately, it turns out that ReLU networks have provable disadvantages in this setting. Hence, we propose to use learnable spline activation functions with at least 3 linear regions instead. We prove that this choice is optimal among all component-wise $1$-Lipschitz activation functions in the sense that no other weight constrained architecture can approximate a larger class of functions. Additionally, this choice is at least as expressive as the recently introduced non component-wise Groupsort activation function for spectral-norm-constrained weights. Previously published numerical results support our theoretical findings.

Related papers

• Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations [52.031701581294804]
  Lipschitz bounds for neural networks can be computed with upper time preservation guarantees. Our paper bridges the gap and extends Lipschitz beyond slope-restricted activation functions. Our proposed analysis is general and provides a unified approach for estimating $ell$ and $ell_infty$ Lipschitz bounds.
  arXiv  Detail & Related papers  (2024-01-25T09:23:31Z)
• 1-Lipschitz Neural Networks are more expressive with N-Activations [19.858602457988194]
  Small changes to a system's inputs should not result in large changes to its outputs. We show that commonly used activation functions, such as MaxMin, unnecessarily restrict the class of representable functions. We introduce the new N-activation function that is provably more expressive than currently popular activation functions.
  arXiv  Detail & Related papers  (2023-11-10T15:12:04Z)
• A Unified Algebraic Perspective on Lipschitz Neural Networks [88.14073994459586]
  This paper introduces a novel perspective unifying various types of 1-Lipschitz neural networks. We show that many existing techniques can be derived and generalized via finding analytical solutions of a common semidefinite programming (SDP) condition. Our approach, called SDP-based Lipschitz Layers (SLL), allows us to design non-trivial yet efficient generalization of convex potential layers.
  arXiv  Detail & Related papers  (2023-03-06T14:31:09Z)
• Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions [14.378778606939665]
  Lipschitz-constrained neural networks have several advantages over unconstrained ones and can be applied to a variety of problems. We show that neural networks with learnable 1-Lipschitz linear splines are known to be more expressive. Our numerical experiments show that our trained networks compare favorably with existing 1-Lipschitz neural architectures.
  arXiv  Detail & Related papers  (2022-10-28T15:56:55Z)
• Rethinking Lipschitz Neural Networks for Certified L-infinity Robustness [33.72713778392896]
  We study certified $ell_infty$ from a novel perspective of representing Boolean functions. We develop a unified Lipschitz network that generalizes prior works, and design a practical version that can be efficiently trained.
  arXiv  Detail & Related papers  (2022-10-04T17:55:27Z)
• Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint [48.25573695787407]
  We prove that large ConvResNets can not only approximate a target function in terms of function value, but also exhibit sufficient first-order smoothness. Our theory partially justifies the benefits of using deep and wide networks in practice.
  arXiv  Detail & Related papers  (2022-06-09T15:35:22Z)
• Training Certifiably Robust Neural Networks with Efficient Local Lipschitz Bounds [99.23098204458336]
  Certified robustness is a desirable property for deep neural networks in safety-critical applications. We show that our method consistently outperforms state-of-the-art methods on MNIST and TinyNet datasets.
  arXiv  Detail & Related papers  (2021-11-02T06:44:10Z)
• Lipschitz Bounded Equilibrium Networks [3.2872586139884623]
  This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. The new parameterization admits a Lipschitz bound during training via unconstrained optimization. In image classification experiments we show that the Lipschitz bounds are very accurate and improve robustness to adversarial attacks.
  arXiv  Detail & Related papers  (2020-10-05T01:00:40Z)
• On Lipschitz Regularization of Convolutional Layers using Toeplitz Matrix Theory [77.18089185140767]
  Lipschitz regularity is established as a key property of modern deep learning. computing the exact value of the Lipschitz constant of a neural network is known to be NP-hard. We introduce a new upper bound for convolutional layers that is both tight and easy to compute.
  arXiv  Detail & Related papers  (2020-06-15T13:23:34Z)
• Lipschitz constant estimation of Neural Networks via sparse polynomial optimization [47.596834444042685]
  LiPopt is a framework for computing increasingly tighter upper bounds on the Lipschitz constant of neural networks. We show how to use the sparse connectivity of a network, to significantly reduce the complexity. We conduct experiments on networks with random weights as well as networks trained on MNIST.
  arXiv  Detail & Related papers  (2020-04-18T18:55:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.