# Frequency learning for structured CNN filters with Gaussian fractional derivatives

arXiv: https://arxiv.org/abs/2111.06660 · HTML: https://mlsanity.com/paper/2111.06660/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 48 references that are in the map.
- First posted on arXiv 12 Nov 2021.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [ShuffleNet: An Extremely Efficient Convolutional Neural Network for Mobile Devices](https://mlsanity.com/paper/1707.01083/) (2017)
- [ShuffleNet V2: Practical Guidelines for Efficient CNN Architecture Design](https://mlsanity.com/paper/1807.11164/) (2018)
- [SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size](https://mlsanity.com/paper/1602.07360/) (2016)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [AMC: AutoML for Model Compression and Acceleration on Mobile Devices](https://mlsanity.com/paper/1802.03494/) (2018)
- [High Frequency Component Helps Explain the Generalization of Convolutional Neural Networks](https://mlsanity.com/paper/1905.13545/) (2019)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [A Fourier Perspective on Model Robustness in Computer Vision](https://mlsanity.com/paper/1906.08988/) (2019)
- [Learning in the Frequency Domain](https://mlsanity.com/paper/2002.12416/) (2020)
- [fastai: A Layered API for Deep Learning](https://mlsanity.com/paper/2002.04688/) (2020)
- [NetAdapt: Platform-Aware Neural Network Adaptation for Mobile Applications](https://mlsanity.com/paper/1804.03230/) (2018)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [SqueezeNext: Hardware-Aware Neural Network Design](https://mlsanity.com/paper/1803.10615/) (2018)
- [Learning Filter Basis for Convolutional Neural Network Compression](https://mlsanity.com/paper/1908.08932/) (2019)
- [Gabor Convolutional Networks](https://mlsanity.com/paper/1705.01450/) (2017)
- [Scaling the Scattering Transform: Deep Hybrid Networks](https://mlsanity.com/paper/1703.08961/) (2017)
- [Structured Receptive Fields in CNNs](https://mlsanity.com/paper/1605.02971/) (2016)
- [Resolution learning in deep convolutional networks using scale-space theory](https://mlsanity.com/paper/2106.03412/) (2021)
- [Scale-covariant and scale-invariant Gaussian derivative networks](https://mlsanity.com/paper/2011.14759/) (2020)
- [Spectral Leakage and Rethinking the Kernel Size in CNNs](https://mlsanity.com/paper/2101.10143/) (2021)
- [Gabor Layers Enhance Network Robustness](https://mlsanity.com/paper/1912.05661/) (2019)

## Built on by (most cited first, 6 of 6)

- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [PDE-CNNs: Axiomatic Derivations and Applications](https://mlsanity.com/paper/2403.15182/) (2024)
- [Frequency Regularization: Restricting Information Redundancy of Convolutional Neural Networks](https://mlsanity.com/paper/2304.07973/) (2023)
- [Approximating Continuous Convolutions for Deep Network Compression](https://mlsanity.com/paper/2210.08951/) (2022)
- [FreqSelect: Frequency-Aware fMRI-to-Image Reconstruction](https://mlsanity.com/paper/2505.12552/) (2025)
- [Modelling and analysis of the 8 filters from the "master key filters hypothesis" for depthwise-separable deep networks in relation to idealized receptive fields based on scale-space theory](https://mlsanity.com/paper/2509.12746/) (2025)

## Abstract (arXiv)

Frequency information lies at the base of discriminating between textures, and therefore between different objects. Classical CNN architectures limit the frequency learning through fixed filter sizes, and lack a way of explicitly controlling it. Here, we build on the structured receptive field filters with Gaussian derivative basis. Yet, rather than using predetermined derivative orders, which typically result in fixed frequency responses for the basis functions, we learn these. We show that by learning the order of the basis we can accurately learn the frequency of the filters, and hence adapt to the optimal frequencies for the underlying learning task. We investigate the well-founded mathematical formulation of fractional derivatives to adapt the filter frequencies during training. Our formulation leads to parameter savings and data efficiency when compared to the standard CNNs and the Gaussian derivative CNN filter networks that we build upon.