# SEIS: Subspace-based Equivariance and Invariance Scores for Neural Representations

arXiv: https://arxiv.org/abs/2602.04054 · HTML: https://mlsanity.com/paper/2602.04054/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 44 references that are in the map.
- First posted on arXiv 3 Feb 2026.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning](https://mlsanity.com/paper/1602.07261/) (2016)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Network Dissection: Quantifying Interpretability of Deep Visual Representations](https://mlsanity.com/paper/1704.05796/) (2017)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [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)
- [On the Benefits of Invariance in Neural Networks](https://mlsanity.com/paper/2005.00178/) (2020)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [On the Strong Correlation Between Model Invariance and Generalization](https://mlsanity.com/paper/2207.07065/) (2022)
- [Understanding the Generalization Benefit of Model Invariance from a Data Perspective](https://mlsanity.com/paper/2111.05529/) (2021)
- [What Affects Learned Equivariance in Deep Image Recognition Models?](https://mlsanity.com/paper/2304.02628/) (2023)
- [In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?](https://mlsanity.com/paper/2210.03773/) (2022)
- [Invariance Measures for Neural Networks](https://mlsanity.com/paper/2310.17404/) (2023)

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

- [Measuring Learned Equivariance: An Information Theoretic Approach](https://mlsanity.com/paper/2610.17269/) (2026)

## Abstract (arXiv)

Understanding how neural representations respond to geometric transformations is essential for evaluating whether learned features preserve meaningful spatial structure. Existing approaches primarily assess robustness primarily by comparing model outputs under transformed inputs, offering limited insight into how geometric information is organized within internal representations and failing to distinguish between information loss and re-encoding. In this work, we introduce SEIS (Subspace-based Equivariance and Invariance Scores), a subspace metric for analyzing layer-wise feature representations under geometric transformations, disentangling equivariance from invariance without requiring labels or explicit knowledge of the transformation. Through controlled experiments across diverse architectures, we uncover several consistent patterns. First, convolutional encoders exhibit a depth-wise transition from strong equivariance to increasing invariance, with both properties stabilizing within the first few training epochs. In segmentation decoders, however, equivariance tends to recover in later layers. Second, this trade-off is not intrinsic but is shaped by training decisions: data augmentation actively strengthens both equivariance and invariance simultaneously, and multi-task learning induces synergistic gains in both properties beyond what either task achieves alone. Extending our analysis beyond convolutional networks, we find that transformer-based models exhibit distinct geometric behaviors, while MLP-Mixers display intermediate characteristics.