# Neural Networks as Paths through the Space of Representations

arXiv: https://arxiv.org/abs/2206.10999 · HTML: https://mlsanity.com/paper/2206.10999/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 46 references that are in the map.
- First posted on arXiv 22 Jun 2022.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Do Wide and Deep Networks Learn the Same Things? Uncovering How Neural Network Representations Vary with Width and Depth](https://mlsanity.com/paper/2010.15327/) (2020)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [The power of deeper networks for expressing natural functions](https://mlsanity.com/paper/1705.05502/) (2017)
- [Neural tuning and representational geometry](https://mlsanity.com/paper/2104.09743/) (2021)
- [Analyzing biological and artificial neural networks: challenges with opportunities for synergy?](https://mlsanity.com/paper/1810.13373/) (2018)
- [Deep Networks from the Principle of Rate Reduction](https://mlsanity.com/paper/2010.14765/) (2020)
- [Geodesic analysis in Kendall's shape space with epidemiological applications](https://mlsanity.com/paper/1906.11950/) (2019)
- [The differential geometry of perceptual similarity](https://mlsanity.com/paper/1708.00138/) (2017)

## Built on by (most cited first, 5 of 5)

- [Collective Model Intelligence Requires Compatible Specialization](https://mlsanity.com/paper/2411.02207/) (2024)
- [Estimating Shape Distances on Neural Representations with Limited Samples](https://mlsanity.com/paper/2310.05742/) (2023)
- [Attentive multilayer fusion for vision transformers](https://mlsanity.com/paper/2601.09322/) (2026)
- [From Robustness to Improved Generalization and Calibration in Pre-trained Language Models](https://mlsanity.com/paper/2404.00758/) (2024)
- [Representation Trajectories Matters: Complementary Evidence for OOD Detection and Image Classification](https://mlsanity.com/paper/2607.26565/) (2026)

## Abstract (arXiv)

Deep neural networks implement a sequence of layer-by-layer operations that are each relatively easy to understand, but the resulting overall computation is generally difficult to understand. We consider a simple hypothesis for interpreting the layer-by-layer construction of useful representations: perhaps the role of each layer is to reformat information to reduce the "distance" to the desired outputs. With this framework, the layer-wise computation implemented by a deep neural network can be viewed as a path through a high-dimensional representation space. We formalize this intuitive idea of a "path" by leveraging recent advances in *metric* representational similarity. We extend existing representational distance methods by computing geodesics, angles, and projections of representations, going beyond mere layer distances. We then demonstrate these tools by visualizing and comparing the paths taken by ResNet and VGG architectures on CIFAR-10. We conclude by sketching additional ways that this kind of representational geometry can be used to understand and interpret network training, and to describe novel kinds of similarities between different models.