# Improving the Sample-Complexity of Deep Classification Networks with Invariant Integration

arXiv: https://arxiv.org/abs/2202.03967 · HTML: https://mlsanity.com/paper/2202.03967/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 63 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 30 of 43)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [BOHB: Robust and Efficient Hyperparameter Optimization at Scale](https://mlsanity.com/paper/1807.01774/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [TI-POOLING: transformation-invariant pooling for feature learning in Convolutional Neural Networks](https://mlsanity.com/paper/1604.06318/) (2016)
- [Attentive Group Equivariant Convolutional Networks](https://mlsanity.com/paper/2002.03830/) (2020)
- [Group Equivariant Stand-Alone Self-Attention For Vision](https://mlsanity.com/paper/2010.00977/) (2020)

## Built on by (most cited first, 7 of 7)

- [Equivariant 3D-Conditional Diffusion Models for Molecular Linker Design](https://mlsanity.com/paper/2210.05274/) (2022)
- [LaneCPP: Continuous 3D Lane Detection using Physical Priors](https://mlsanity.com/paper/2406.08381/) (2024)
- [Interspace Pruning: Using Adaptive Filter Representations to Improve Training of Sparse CNNs](https://mlsanity.com/paper/2203.07808/) (2022)
- [Dimensionality Reduced Training by Pruning and Freezing Parts of a Deep Neural Network, a Survey](https://mlsanity.com/paper/2205.08099/) (2022)
- [Pioneering SE(2)-Equivariant Trajectory Planning for Automated Driving](https://mlsanity.com/paper/2403.11304/) (2024)
- [Deep Neural Networks with Efficient Guaranteed Invariances](https://mlsanity.com/paper/2303.01567/) (2023)
- [Domain Generalization In Robust Invariant Representation](https://mlsanity.com/paper/2304.03431/) (2023)

## Abstract (arXiv)

Leveraging prior knowledge on intraclass variance due to transformations is a powerful method to improve the sample complexity of deep neural networks. This makes them applicable to practically important use-cases where training data is scarce. Rather than being learned, this knowledge can be embedded by enforcing invariance to those transformations. Invariance can be imposed using group-equivariant convolutions followed by a pooling operation. For rotation-invariance, previous work investigated replacing the spatial pooling operation with invariant integration which explicitly constructs invariant representations. Invariant integration uses monomials which are selected using an iterative approach requiring expensive pre-training. We propose a novel monomial selection algorithm based on pruning methods to allow an application to more complex problems. Additionally, we replace monomials with different functions such as weighted sums, multi-layer perceptrons and self-attention, thereby streamlining the training of invariant-integration-based architectures. We demonstrate the improved sample complexity on the Rotated-MNIST, SVHN and CIFAR-10 datasets where rotation-invariant-integration-based Wide-ResNet architectures using monomials and weighted sums outperform the respective baselines in the limited sample regime. We achieve state-of-the-art results using full data on Rotated-MNIST and SVHN where rotation is a main source of intraclass variation. On STL-10 we outperform a standard and a rotation-equivariant convolutional neural network using pooling.