# Laplacian Networks: Bounding Indicator Function Smoothness for Neural Network Robustness

arXiv: https://arxiv.org/abs/1805.10133 · HTML: https://mlsanity.com/paper/1805.10133/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2018.

## Built on by (most cited first, 7 of 7)

- [A Survey on the Robustness of Computer Vision Models against Common Corruptions](https://mlsanity.com/paper/2305.06024/) (2023)
- [Channel-Wise Early Stopping without a Validation Set via NNK Polytope Interpolation](https://mlsanity.com/paper/2107.12972/) (2021)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Channel redundancy and overlap in convolutional neural networks with channel-wise NNK graphs](https://mlsanity.com/paper/2110.11400/) (2021)
- [Frequency Dropout: Feature-Level Regularization via Randomized Filtering](https://mlsanity.com/paper/2209.09844/) (2022)
- [Study of Manifold Geometry using Multiscale Non-Negative Kernel Graphs](https://mlsanity.com/paper/2210.17475/) (2022)
- [Graphs as Tools to Improve Deep Learning Methods](https://mlsanity.com/paper/2110.03999/) (2021)

## Abstract (arXiv)

For the past few years, Deep Neural Network (DNN) robustness has become a question of paramount importance. As a matter of fact, in sensitive settings misclassification can lead to dramatic consequences. Such misclassifications are likely to occur when facing adversarial attacks, hardware failures or limitations, and imperfect signal acquisition. To address this question, authors have proposed different approaches aiming at increasing the robustness of DNNs, such as adding regularizers or training using noisy examples. In this paper we propose a new regularizer built upon the Laplacian of similarity graphs obtained from the representation of training data at each layer of the DNN architecture. This regularizer penalizes large changes (across consecutive layers in the architecture) in the distance between examples of different classes, and as such enforces smooth variations of the class boundaries. Since it is agnostic to the type of deformations that are expected when predicting with the DNN, the proposed regularizer can be combined with existing ad-hoc methods. We provide theoretical justification for this regularizer and demonstrate its effectiveness to improve robustness of DNNs on classical supervised learning vision datasets.