# Duality of Bures and Shape Distances with Implications for Comparing Neural Representations

arXiv: https://arxiv.org/abs/2311.11436 · HTML: https://mlsanity.com/paper/2311.11436/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 57 references that are in the map.
- First posted on arXiv 19 Nov 2023.

## Builds on (most cited first, 19 of 19)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [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)
- [Convergent Learning: Do different neural networks learn the same representations?](https://mlsanity.com/paper/1511.07543/) (2015)
- [Getting aligned on representational alignment](https://mlsanity.com/paper/2310.13018/) (2023)
- [Similarity of Neural Network Models: A Survey of Functional and Representational Measures](https://mlsanity.com/paper/2305.06329/) (2023)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Reliability of CKA as a Similarity Measure in Deep Learning](https://mlsanity.com/paper/2210.16156/) (2022)
- [Universality and individuality in neural dynamics across large populations of recurrent networks](https://mlsanity.com/paper/1907.08549/) (2019)
- [Recurrence is required to capture the representational dynamics of the human visual system](https://mlsanity.com/paper/1903.05946/) (2019)
- [Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions](https://mlsanity.com/paper/1805.07594/) (2018)
- [Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes](https://mlsanity.com/paper/1801.01990/) (2018)
- [Representational dissimilarity metric spaces for stochastic neural networks](https://mlsanity.com/paper/2211.11665/) (2022)
- [Similarity of Neural Networks with Gradients](https://mlsanity.com/paper/2003.11498/) (2020)
- [Statistical inference on representational geometries](https://mlsanity.com/paper/2112.09200/) (2021)
- [Inferring brain-computational mechanisms with models of activity measurements](https://mlsanity.com/paper/1608.02027/) (2016)

## Built on by (most cited first, 15 of 15)

- [From Language to Cognition: How LLMs Outgrow the Human Language Network](https://mlsanity.com/paper/2503.01830/) (2025)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [What Representational Similarity Measures Imply about Decodable Information](https://mlsanity.com/paper/2411.08197/) (2024)
- [Differentiable Optimization of Similarity Scores Between Models and Brains](https://mlsanity.com/paper/2407.07059/) (2024)
- [On the universality of neural encodings in CNNs](https://mlsanity.com/paper/2409.19460/) (2024)
- [Brain-Like Language Processing via a Shallow Untrained Multihead Attention Network](https://mlsanity.com/paper/2406.15109/) (2024)
- [Model-brain comparison using inter-animal transforms](https://mlsanity.com/paper/2510.02523/) (2025)
- [Bayesian Comparisons Between Representations](https://mlsanity.com/paper/2411.08739/) (2024)
- [Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example](https://mlsanity.com/paper/2506.06904/) (2025)
- [Modulating Cross-Modal Convergence with Single-Stimulus, Intra-Modal Dispersion](https://mlsanity.com/paper/2604.21836/) (2026)
- [How Data Augmentation Shapes Neural Representations](https://mlsanity.com/paper/2605.15306/) (2026)
- [Supervised Quadratic Feature Analysis: Information Geometry Approach for Dimensionality Reduction](https://mlsanity.com/paper/2502.00168/) (2025)
- [Knowledge distillation through geometry-aware representational alignment](https://mlsanity.com/paper/2509.25253/) (2025)
- [Sparse components distinguish visual pathways & their alignment to neural networks](https://mlsanity.com/paper/2510.08858/) (2025)
- [Scale, Drivers, and Limits of Representational Alignment Across Large Language Models](https://mlsanity.com/paper/2610.14476/) (2026)

## Abstract (arXiv)

A multitude of (dis)similarity measures between neural network representations have been proposed, resulting in a fragmented research landscape. Most of these measures fall into one of two categories. First, measures such as linear regression, canonical correlations analysis (CCA), and shape distances, all learn explicit mappings between neural units to quantify similarity while accounting for expected invariances. Second, measures such as representational similarity analysis (RSA), centered kernel alignment (CKA), and normalized Bures similarity (NBS) all quantify similarity in summary statistics, such as stimulus-by-stimulus kernel matrices, which are already invariant to expected symmetries. Here, we take steps towards unifying these two broad categories of methods by observing that the cosine of the Riemannian shape distance (from category 1) is equal to NBS (from category 2). We explore how this connection leads to new interpretations of shape distances and NBS, and draw contrasts of these measures with CKA, a popular similarity measure in the deep learning literature.