# Non-linear Convolution Filters for CNN-based Learning

arXiv: https://arxiv.org/abs/1708.07038 · HTML: https://mlsanity.com/paper/1708.07038/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Aug 2017.

## Built on by (most cited first, 30 of 38)

- [B-cos Networks: Alignment is All We Need for Interpretability](https://mlsanity.com/paper/2205.10268/) (2022)
- [B-cos Alignment for Inherently Interpretable CNNs and Vision Transformers](https://mlsanity.com/paper/2306.10898/) (2023)
- [Resolution learning in deep convolutional networks using scale-space theory](https://mlsanity.com/paper/2106.03412/) (2021)
- [StEik: Stabilizing the Optimization of Neural Signed Distance Functions and Finer Shape Representation](https://mlsanity.com/paper/2305.18414/) (2023)
- [QuadraLib: A Performant Quadratic Neural Network Library for Architecture Optimization and Design Exploration](https://mlsanity.com/paper/2204.01701/) (2022)
- [On Expressivity and Trainability of Quadratic Networks](https://mlsanity.com/paper/2110.06081/) (2021)
- [Surpassing the Human Accuracy: Detecting Gallbladder Cancer from USG Images with Curriculum Learning](https://mlsanity.com/paper/2204.11433/) (2022)
- [On-Off Center-Surround Receptive Fields for Accurate and Robust Image Classification](https://mlsanity.com/paper/2106.07091/) (2021)
- [VolterraNet: A higher order convolutional network with group equivariance for homogeneous manifolds](https://mlsanity.com/paper/2106.15301/) (2021)
- [DeViL: Decoding Vision features into Language](https://mlsanity.com/paper/2309.01617/) (2023)
- [Towards NeuroAI: Introducing Neuronal Diversity into Artificial Neural Networks](https://mlsanity.com/paper/2301.09245/) (2023)
- [One Neuron Saved Is One Neuron Earned: On Parametric Efficiency of Quadratic Networks](https://mlsanity.com/paper/2303.06316/) (2023)
- [Quadratic Neuron-empowered Heterogeneous Autoencoder for Unsupervised Anomaly Detection](https://mlsanity.com/paper/2204.01707/) (2022)
- [Neural Echos: Depthwise Convolutional Filters Replicate Biological Receptive Fields](https://mlsanity.com/paper/2401.10178/) (2024)
- [Convolution goes higher-order: a biologically inspired mechanism empowers image classification](https://mlsanity.com/paper/2412.06740/) (2024)
- [Deep Unrolling Networks with Recurrent Momentum Acceleration for Nonlinear Inverse Problems](https://mlsanity.com/paper/2307.16120/) (2023)
- [No One-Size-Fits-All Neurons: Task-based Neurons for Artificial Neural Networks](https://mlsanity.com/paper/2405.02369/) (2024)
- [Convolutional versus Self-Organized Operational Neural Networks for Real-World Blind Image Denoising](https://mlsanity.com/paper/2103.03070/) (2021)
- [Latent Code-Based Fusion: A Volterra Neural Network Approach](https://mlsanity.com/paper/2104.04829/) (2021)
- [Towards Understanding the Effectiveness of Attention Mechanism](https://mlsanity.com/paper/2106.15067/) (2021)
- [Kernel function impact on convolutional neural networks](https://mlsanity.com/paper/2302.10266/) (2023)
- [Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications](https://mlsanity.com/paper/2305.16657/) (2023)
- [Single Domain Generalization via Normalised Cross-correlation Based Convolutions](https://mlsanity.com/paper/2307.05901/) (2023)
- [STENCIL-NET: Data-driven solution-adaptive discretization of partial differential equations](https://mlsanity.com/paper/2101.06182/) (2021)
- [NL-CNN: A Resources-Constrained Deep Learning Model based on Nonlinear Convolution](https://mlsanity.com/paper/2102.00227/) (2021)
- [Brain-inspired algorithms for processing of visual data](https://mlsanity.com/paper/2103.01634/) (2021)
- [Cloud-RAIN: Point Cloud Analysis with Reflectional Invariance](https://mlsanity.com/paper/2305.07814/) (2023)
- [Computational and Storage Efficient Quadratic Neurons for Deep Neural Networks](https://mlsanity.com/paper/2306.07294/) (2023)
- [Efficient Visual Representation Learning with Heat Conduction Equation](https://mlsanity.com/paper/2408.05901/) (2024)
- [IC Networks: Remodeling the Basic Unit for Convolutional Neural Networks](https://mlsanity.com/paper/2102.03495/) (2021)

## Abstract (arXiv)

During the last years, Convolutional Neural Networks (CNNs) have achieved state-of-the-art performance in image classification. Their architectures have largely drawn inspiration by models of the primate visual system. However, while recent research results of neuroscience prove the existence of non-linear operations in the response of complex visual cells, little effort has been devoted to extend the convolution technique to non-linear forms. Typical convolutional layers are linear systems, hence their expressiveness is limited. To overcome this, various non-linearities have been used as activation functions inside CNNs, while also many pooling strategies have been applied. We address the issue of developing a convolution method in the context of a computational model of the visual cortex, exploring quadratic forms through the Volterra kernels. Such forms, constituting a more rich function space, are used as approximations of the response profile of visual cells. Our proposed second-order convolution is tested on CIFAR-10 and CIFAR-100. We show that a network which combines linear and non-linear filters in its convolutional layers, can outperform networks that use standard linear filters with the same architecture, yielding results competitive with the state-of-the-art on these datasets.