# Generalizing the Convolution Operator in Convolutional Neural Networks

arXiv: https://arxiv.org/abs/1707.09864 · HTML: https://mlsanity.com/paper/1707.09864/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2017.

## Built on by (most cited first, 9 of 9)

- [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)
- [Pixel-Grounded Prototypical Part Networks](https://mlsanity.com/paper/2309.14531/) (2023)
- [Single Domain Generalization via Normalised Cross-correlation Based Convolutions](https://mlsanity.com/paper/2307.05901/) (2023)
- [A Robust Prototype-Based Network with Interpretable RBF Classifier Foundations](https://mlsanity.com/paper/2412.15499/) (2024)
- [Optimal Density Functions for Weighted Convolution in Learning Models](https://mlsanity.com/paper/2505.24527/) (2025)
- [Optimal Weighted Convolution for Classification and Denosing](https://mlsanity.com/paper/2505.24558/) (2025)
- [Predicting Human Trajectories by Learning and Matching Patterns](https://mlsanity.com/paper/2104.10241/) (2021)
- [PowerLinear Activation Functions with application to the first layer of CNNs](https://mlsanity.com/paper/2108.09256/) (2021)

## Abstract (arXiv)

Convolutional neural networks have become a main tool for solving many machine vision and machine learning problems. A major element of these networks is the convolution operator which essentially computes the inner product between a weight vector and the vectorized image patches extracted by sliding a window in the image planes of the previous layer. In this paper, we propose two classes of surrogate functions for the inner product operation inherent in the convolution operator and so attain two generalizations of the convolution operator. The first one is the class of positive definite kernel functions where their application is justified by the kernel trick. The second one is the class of similarity measures defined based on a distance function. We justify this by tracing back to the basic idea behind the neocognitron which is the ancestor of CNNs. Both methods are then further generalized by allowing a monotonically increasing function to be applied subsequently. Like any trainable parameter in a neural network, the template pattern and the parameters of the kernel/distance function are trained with the back-propagation algorithm. As an aside, we use the proposed framework to justify the use of sine activation function in CNNs. Our experiments on the MNIST dataset show that the performance of ordinary CNNs can be achieved by generalized CNNs based on weighted L1/L2 distances, proving the applicability of the proposed generalization of the convolutional neural networks.