# Invariance Measures for Neural Networks

arXiv: https://arxiv.org/abs/2310.17404 · HTML: https://mlsanity.com/paper/2310.17404/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 46 references that are in the map.
- First posted on arXiv 26 Oct 2023.

## Builds on (most cited first, 23 of 23)

- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [Controllable Invariance through Adversarial Feature Learning](https://mlsanity.com/paper/1705.11122/) (2017)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Understanding and Improving Convolutional Neural Networks via Concatenated Rectified Linear Units](https://mlsanity.com/paper/1603.05201/) (2016)
- [Exploring the Landscape of Spatial Robustness](https://mlsanity.com/paper/1712.02779/) (2017)
- [Fundamental Tradeoffs between Invariance and Sensitivity to Adversarial Perturbations](https://mlsanity.com/paper/2002.04599/) (2020)
- [NullHop: A Flexible Convolutional Neural Network Accelerator Based on Sparse Representations of Feature Maps](https://mlsanity.com/paper/1706.01406/) (2017)
- [Quantifying Translation-Invariance in Convolutional Neural Networks](https://mlsanity.com/paper/1801.01450/) (2017)
- [Manitest: Are classifiers really invariant?](https://mlsanity.com/paper/1507.06535/) (2015)
- [Revisiting Data Augmentation for Rotational Invariance in Convolutional Neural Networks](https://mlsanity.com/paper/2310.08429/) (2023)
- [Sample-Optimal Low-Rank Approximation of Distance Matrices](https://mlsanity.com/paper/1906.00339/) (2019)
- [Towards Understanding the Invertibility of Convolutional Neural Networks](https://mlsanity.com/paper/1705.08664/) (2017)
- [Diverse feature visualizations reveal invariances in early layers of deep neural networks](https://mlsanity.com/paper/1807.10589/) (2018)
- [Explaining the Unexplained: A CLass-Enhanced Attentive Response (CLEAR) Approach to Understanding Deep Neural Networks](https://mlsanity.com/paper/1704.04133/) (2017)
- [Studying Invariances of Trained Convolutional Neural Networks](https://mlsanity.com/paper/1803.05963/) (2018)
- [Learning Deep Object Detectors from 3D Models](https://mlsanity.com/paper/1412.7122/) (2014)
- [3D Human Pose Estimation with Siamese Equivariant Embedding](https://mlsanity.com/paper/1809.07217/) (2018)
- [The Effect of Learning Strategy versus Inherent Architecture Properties on the Ability of Convolutional Neural Networks to Develop Transformation Invariance](https://mlsanity.com/paper/1810.13128/) (2018)

## Built on by (most cited first, 1 of 1)

- [SEIS: Subspace-based Equivariance and Invariance Scores for Neural Representations](https://mlsanity.com/paper/2602.04054/) (2026)

## Abstract (arXiv)

Invariances in neural networks are useful and necessary for many tasks. However, the representation of the invariance of most neural network models has not been characterized. We propose measures to quantify the invariance of neural networks in terms of their internal representation. The measures are efficient and interpretable, and can be applied to any neural network model. They are also more sensitive to invariance than previously defined measures. We validate the measures and their properties in the domain of affine transformations and the CIFAR10 and MNIST datasets, including their stability and interpretability. Using the measures, we perform a first analysis of CNN models and show that their internal invariance is remarkably stable to random weight initializations, but not to changes in dataset or transformation. We believe the measures will enable new avenues of research in invariance representation.