# Warped Convolutions: Efficient Invariance to Spatial Transformations

arXiv: https://arxiv.org/abs/1609.04382 · HTML: https://mlsanity.com/paper/1609.04382/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2016.

## Built on by (most cited first, 24 of 24)

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- [RIC-CNN: Rotation-Invariant Coordinate Convolutional Neural Network](https://mlsanity.com/paper/2211.11812/) (2022)
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- [Training or Architecture? How to Incorporate Invariance in Neural Networks](https://mlsanity.com/paper/2106.10044/) (2021)
- [Homography Decomposition Networks for Planar Object Tracking](https://mlsanity.com/paper/2112.07909/) (2021)
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- [Rethinking Rotation in Self-Supervised Contrastive Learning: Adaptive Positive or Negative Data Augmentation](https://mlsanity.com/paper/2210.12681/) (2022)
- [The Devils in the Point Clouds: Studying the Robustness of Point Cloud Convolutions](https://mlsanity.com/paper/2101.07832/) (2021)
- [A Simple Strategy to Provable Invariance via Orbit Mapping](https://mlsanity.com/paper/2209.11916/) (2022)
- [RePoseDM: Recurrent Pose Alignment and Gradient Guidance for Pose Guided Image Synthesis](https://mlsanity.com/paper/2310.16074/) (2023)
- [FILTRA: Rethinking Steerable CNN by Filter Transform](https://mlsanity.com/paper/2105.11636/) (2021)
- [Quantised Transforming Auto-Encoders: Achieving Equivariance to Arbitrary Transformations in Deep Networks](https://mlsanity.com/paper/2111.12873/) (2021)
- [Achieving Rotation-Invariant Convolution via Non-Learnable Orientation Alignment Operators](https://mlsanity.com/paper/2404.11309/) (2024)
- [Warped Convolutional Networks: Bridge Homography to sl(3) algebra by Group Convolution](https://mlsanity.com/paper/2206.11657/) (2022)
- [Subspace-Configurable Networks](https://mlsanity.com/paper/2305.13536/) (2023)
- [Restore Translation Using Equivariant Neural Networks](https://mlsanity.com/paper/2306.16938/) (2023)
- [Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold](https://mlsanity.com/paper/2307.12517/) (2023)
- [MARs: Multi-view Attention Regularizations for Patch-based Feature Recognition of Space Terrain](https://mlsanity.com/paper/2410.05182/) (2024)
- [HiResNets: Native Full-HD Video Recognition with Foveal Residual Streams](https://mlsanity.com/paper/2608.02140/) (2026)

## Abstract (arXiv)

Convolutional Neural Networks (CNNs) are extremely efficient, since they exploit the inherent translation-invariance of natural images. However, translation is just one of a myriad of useful spatial transformations. Can the same efficiency be attained when considering other spatial invariances? Such generalized convolutions have been considered in the past, but at a high computational cost. We present a construction that is simple and exact, yet has the same computational complexity that standard convolutions enjoy. It consists of a constant image warp followed by a simple convolution, which are standard blocks in deep learning toolboxes. With a carefully crafted warp, the resulting architecture can be made equivariant to a wide range of two-parameter spatial transformations. We show encouraging results in realistic scenarios, including the estimation of vehicle poses in the Google Earth dataset (rotation and scale), and face poses in Annotated Facial Landmarks in the Wild (3D rotations under perspective).