# Soft Matching Distance: A metric on neural representations that captures single-neuron tuning

arXiv: https://arxiv.org/abs/2311.09466 · HTML: https://mlsanity.com/paper/2311.09466/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 51 references that are in the map.
- First posted on arXiv 16 Nov 2023.

## Builds on (most cited first, 14 of 14)

- [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)
- [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)
- [Convergent Learning: Do different neural networks learn the same representations?](https://mlsanity.com/paper/1511.07543/) (2015)
- [Model Fusion via Optimal Transport](https://mlsanity.com/paper/1910.05653/) (2019)
- [Object Detectors Emerge in Deep Scene CNNs](https://mlsanity.com/paper/1412.6856/) (2014)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [On the importance of single directions for generalization](https://mlsanity.com/paper/1803.06959/) (2018)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Towards Optimal Transport with Global Invariances](https://mlsanity.com/paper/1806.09277/) (2018)
- [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)
- [Representation Transfer by Optimal Transport](https://mlsanity.com/paper/2007.06737/) (2020)

## Built on by (most cited first, 18 of 18)

- [Learned feature representations are biased by complexity, learning order, position, and more](https://mlsanity.com/paper/2405.05847/) (2024)
- [What Representational Similarity Measures Imply about Decodable Information](https://mlsanity.com/paper/2411.08197/) (2024)
- [Brain-Model Evaluations Need the NeuroAI Turing Test](https://mlsanity.com/paper/2502.16238/) (2025)
- [Evaluating Representational Similarity Measures from the Lens of Functional Correspondence](https://mlsanity.com/paper/2411.14633/) (2024)
- [Bridging Critical Gaps in Convergent Learning: How Representational Alignment Evolves Across Layers, Training, and Distribution Shifts](https://mlsanity.com/paper/2502.18710/) (2025)
- [Representation biases: will we achieve complete understanding by analyzing representations?](https://mlsanity.com/paper/2507.22216/) (2025)
- [Superposition disentanglement of neural representations reveals hidden alignment](https://mlsanity.com/paper/2510.03186/) (2025)
- [Model Alignment Search](https://mlsanity.com/paper/2501.06164/) (2025)
- [Representational Alignment Across Model Layers and Brain Regions with Multi-Level Optimal Transport](https://mlsanity.com/paper/2510.01706/) (2025)
- [Model-brain comparison using inter-animal transforms](https://mlsanity.com/paper/2510.02523/) (2025)
- [Measuring the Measures: Discriminative Capacity of Representational Similarity Metrics Across Model Families](https://mlsanity.com/paper/2509.04622/) (2025)
- [Seeing Through Words, Speaking Through Pixels: Deep Representational Alignment Between Vision and Language Models](https://mlsanity.com/paper/2509.20751/) (2025)
- [Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks](https://mlsanity.com/paper/2607.08561/) (2026)
- [Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example](https://mlsanity.com/paper/2506.06904/) (2025)
- [Comparing and Integrating Different Notions of Representational Correspondence in Neural Systems](https://mlsanity.com/paper/2509.21628/) (2025)
- [Integrated representational signatures strengthen specificity in brains and models](https://mlsanity.com/paper/2510.20847/) (2025)
- [Partial Soft-Matching Distance for Neural Representational Comparison with Partial Unit Correspondence](https://mlsanity.com/paper/2602.19331/) (2026)
- [Scale, Drivers, and Limits of Representational Alignment Across Large Language Models](https://mlsanity.com/paper/2610.14476/) (2026)

## Abstract (arXiv)

Common measures of neural representational (dis)similarity are designed to be insensitive to rotations and reflections of the neural activation space. Motivated by the premise that the tuning of individual units may be important, there has been recent interest in developing stricter notions of representational (dis)similarity that require neurons to be individually matched across networks. When two networks have the same size (i.e. same number of neurons), a distance metric can be formulated by optimizing over neuron index permutations to maximize tuning curve alignment. However, it is not clear how to generalize this metric to measure distances between networks with different sizes. Here, we leverage a connection to optimal transport theory to derive a natural generalization based on "soft" permutations. The resulting metric is symmetric, satisfies the triangle inequality, and can be interpreted as a Wasserstein distance between two empirical distributions. Further, our proposed metric avoids counter-intuitive outcomes suffered by alternative approaches, and captures complementary geometric insights into neural representations that are entirely missed by rotation-invariant metrics.