# Scale equivariance in CNNs with vector fields

arXiv: https://arxiv.org/abs/1807.11783 · HTML: https://mlsanity.com/paper/1807.11783/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2018.

## Built on by (most cited first, 20 of 20)

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- [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)
- [Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views?](https://mlsanity.com/paper/2110.07472/) (2021)
- [Just a Matter of Scale? Reevaluating Scale Equivariance in Convolutional Neural Networks](https://mlsanity.com/paper/2211.10288/) (2022)
- [Deep Neural Networks with Efficient Guaranteed Invariances](https://mlsanity.com/paper/2303.01567/) (2023)
- [Adaptive aggregation of Monte Carlo augmented decomposed filters for efficient group-equivariant convolutional neural network](https://mlsanity.com/paper/2305.10110/) (2023)
- [A study on the invariance in security whatever the dimension of images for the steganalysis by deep-learning](https://mlsanity.com/paper/2302.11527/) (2023)
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- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)

## Abstract (arXiv)

We study the effect of injecting local scale equivariance into Convolutional Neural Networks. This is done by applying each convolutional filter at multiple scales. The output is a vector field encoding for the maximally activating scale and the scale itself, which is further processed by the following convolutional layers. This allows all the intermediate representations to be locally scale equivariant. We show that this improves the performance of the model by over 20% in the scale equivariant task of regressing the scaling factor applied to randomly scaled MNIST digits. Furthermore, we find it also useful for scale invariant tasks, such as the actual classification of randomly scaled digits. This highlights the usefulness of allowing for a compact representation that can also learn relationships between different local scales by keeping internal scale equivariance.