# The Topology and Geometry of Neural Representations

arXiv: https://arxiv.org/abs/2309.11028 · HTML: https://mlsanity.com/paper/2309.11028/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 104 references that are in the map.
- First posted on arXiv 20 Sep 2023.

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

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Recurrence is required to capture the representational dynamics of the human visual system](https://mlsanity.com/paper/1903.05946/) (2019)
- [Neural population geometry: An approach for understanding biological and artificial neural networks](https://mlsanity.com/paper/2104.07059/) (2021)
- [Deep Neural Networks Rival the Representation of Primate IT Cortex for Core Visual Object Recognition](https://mlsanity.com/paper/1406.3284/) (2014)
- [Cognitive computational neuroscience](https://mlsanity.com/paper/1807.11819/) (2018)
- [Neural tuning and representational geometry](https://mlsanity.com/paper/2104.09743/) (2021)
- [Topological Data Analysis](https://mlsanity.com/paper/1609.08227/) (2016)
- [Statistical inference on representational geometries](https://mlsanity.com/paper/2112.09200/) (2021)
- [Controllability of Brain Networks](https://mlsanity.com/paper/1406.5197/) (2014)
- [Models of communication and control for brain networks: distinctions, convergence, and future outlook](https://mlsanity.com/paper/2002.07029/) (2020)
- [Geometric and Topological Inference for Deep Representations of Complex Networks](https://mlsanity.com/paper/2203.05488/) (2022)
- [Visualizing Representational Dynamics with Multidimensional Scaling Alignment](https://mlsanity.com/paper/1906.09264/) (2019)
- [Brain and Cognitive Reserve: Translation via Network Control Theory](https://mlsanity.com/paper/1604.04683/) (2016)
- [Topological Data Analysis in Time Series: Temporal Filtration and Application to Single-Cell Genomics](https://mlsanity.com/paper/2204.14048/) (2022)

## Built on by (most cited first, 14 of 14)

- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [Comparing the information content of probabilistic representation spaces](https://mlsanity.com/paper/2405.21042/) (2024)
- [Geometric Stability: The Missing Axis of Representations](https://mlsanity.com/paper/2601.09173/) (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)
- [Platonic Representations in the Human Brain: Unsupervised Recovery of Universal Geometry](https://mlsanity.com/paper/2605.20496/) (2026)
- [SoftStep: Learning Sparse Similarity Powers Deep Neighbor-Based Regression](https://mlsanity.com/paper/2506.08139/) (2025)
- [Algorithm Development in Neural Networks: Insights from the Streaming Parity Task](https://mlsanity.com/paper/2507.09897/) (2025)
- [Universal Latent Homeomorphic Manifolds: A Framework for Cross-Domain Representation Unification](https://mlsanity.com/paper/2601.09025/) (2026)
- [BrainDyn: A Sheaf Neural ODE for Forecasting and Generating Brain Dynamics](https://mlsanity.com/paper/2605.19324/) (2026)
- [MAPS: A Synthetic Dataset for Probing Vision Models in a Controlled 3D Scene Space](https://mlsanity.com/paper/2605.20549/) (2026)
- [Context Is King: How In-Context Specification Shapes the Geometry of Concepts](https://mlsanity.com/paper/2607.24425/) (2026)
- [TAGTorch: A PyTorch Library for Geometry, Topology, and Symmetry-Aware Machine Learning](https://mlsanity.com/paper/2607.28755/) (2026)
- [Geometric Similarity in VLM Low-Level Vision Representations](https://mlsanity.com/paper/2610.00848/) (2026)
- [Scale, Drivers, and Limits of Representational Alignment Across Large Language Models](https://mlsanity.com/paper/2610.14476/) (2026)

## Abstract (arXiv)

A central question for neuroscience is how to characterize brain representations of perceptual and cognitive content. An ideal characterization should distinguish different functional regions with robustness to noise and idiosyncrasies of individual brains that do not correspond to computational differences. Previous studies have characterized brain representations by their representational geometry, which is defined by the representational dissimilarity matrix (RDM), a summary statistic that abstracts from the roles of individual neurons (or responses channels) and characterizes the discriminability of stimuli. Here we explore a further step of abstraction: from the geometry to the topology of brain representations. We propose topological representational similarity analysis (tRSA), an extension of representational similarity analysis (RSA) that uses a family of geo-topological summary statistics that generalizes the RDM to characterize the topology while de-emphasizing the geometry. We evaluate this new family of statistics in terms of the sensitivity and specificity for model selection using both simulations and fMRI data. In the simulations, the ground truth is a data-generating layer representation in a neural network model and the models are the same and other layers in different model instances (trained from different random seeds). In fMRI, the ground truth is a visual area and the models are the same and other areas measured in different subjects. Results show that topology-sensitive characterizations of population codes are robust to noise and interindividual variability and maintain excellent sensitivity to the unique representational signatures of different neural network layers and brain regions. These methods enable researchers to calibrate comparisons among representations in brains and models to be sensitive to the geometry, the topology, or a combination of both.