# Neural population geometry: An approach for understanding biological and artificial neural networks

arXiv: https://arxiv.org/abs/2104.07059 · HTML: https://mlsanity.com/paper/2104.07059/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 67 references that are in the map.
- First posted on arXiv 14 Apr 2021.

## Builds on (most cited first, 8 of 8)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [Emergence of Separable Manifolds in Deep Language Representations](https://mlsanity.com/paper/2006.01095/) (2020)
- [Analyzing biological and artificial neural networks: challenges with opportunities for synergy?](https://mlsanity.com/paper/1810.13373/) (2018)
- [Visualizing the PHATE of Neural Networks](https://mlsanity.com/paper/1908.02831/) (2019)
- [Clique topology reveals intrinsic geometric structure in neural correlations](https://mlsanity.com/paper/1502.06172/) (2015)
- [Untangling in Invariant Speech Recognition](https://mlsanity.com/paper/2003.01787/) (2020)
- [Linear Readout of Object Manifolds](https://mlsanity.com/paper/1512.01834/) (2015)

## Built on by (most cited first, 30 of 52)

- [Large language models implicitly learn to straighten neural sentence trajectories to construct a predictive representation of natural language](https://mlsanity.com/paper/2311.04930/) (2023)
- [Learning Efficient Coding of Natural Images with Maximum Manifold Capacity Representations](https://mlsanity.com/paper/2303.03307/) (2023)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Tensor-CSPNet: A Novel Geometric Deep Learning Framework for Motor Imagery Classification](https://mlsanity.com/paper/2202.02472/) (2022)
- [The Geometry of Prompting: Unveiling Distinct Mechanisms of Task Adaptation in Language Models](https://mlsanity.com/paper/2502.08009/) (2025)
- [Probing artificial neural networks: insights from neuroscience](https://mlsanity.com/paper/2104.08197/) (2021)
- [Representational dissimilarity metric spaces for stochastic neural networks](https://mlsanity.com/paper/2211.11665/) (2022)
- [Operationalising Representation in Natural Language Processing](https://mlsanity.com/paper/2306.08193/) (2023)
- [The Topology and Geometry of Neural Representations](https://mlsanity.com/paper/2309.11028/) (2023)
- [A Spectral Theory of Neural Prediction and Alignment](https://mlsanity.com/paper/2309.12821/) (2023)
- [Neural Population Geometry Reveals the Role of Stochasticity in Robust Perception](https://mlsanity.com/paper/2111.06979/) (2021)
- [Are Deep Neural Networks Adequate Behavioural Models of Human Visual Perception?](https://mlsanity.com/paper/2305.17023/) (2023)
- [Linear Classification of Neural Manifolds with Correlated Variability](https://mlsanity.com/paper/2211.14961/) (2022)
- [Neural population geometry and optimal coding of tasks with shared latent structure](https://mlsanity.com/paper/2402.16770/) (2024)
- [Feature Learning beyond the Lazy-Rich Dichotomy: Insights from Representational Geometry](https://mlsanity.com/paper/2503.18114/) (2025)
- [Discriminating image representations with principal distortions](https://mlsanity.com/paper/2410.15433/) (2024)
- [On convex decision regions in deep network representations](https://mlsanity.com/paper/2305.17154/) (2023)
- [Multilevel Interpretability Of Artificial Neural Networks: Leveraging Framework And Methods From Neuroscience](https://mlsanity.com/paper/2408.12664/) (2024)
- [Bi-Orthogonal Factor Decomposition for Vision Transformers](https://mlsanity.com/paper/2601.05328/) (2026)
- [Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds](https://mlsanity.com/paper/2606.25234/) (2026)
- [Probing Biological and Artificial Neural Networks with Task-dependent Neural Manifolds](https://mlsanity.com/paper/2312.14285/) (2023)
- [Emergent Riemannian geometry over learning discrete computations on continuous manifolds](https://mlsanity.com/paper/2512.00196/) (2025)
- [A Geometric Unification of Concept Learning with Concept Cones](https://mlsanity.com/paper/2512.07355/) (2025)
- [Soft-margin classification of object manifolds](https://mlsanity.com/paper/2203.07040/) (2022)
- [Nonlinear classification of neural manifolds with contextual information](https://mlsanity.com/paper/2405.06851/) (2024)
- [Diagnosing Generalization Failures from Representational Geometry Markers](https://mlsanity.com/paper/2603.01879/) (2026)
- [A Compositional Framework for Open-ended Intelligence](https://mlsanity.com/paper/2606.15386/) (2026)
- [Language Models Represent and Transform Concepts with Shared Geometry](https://mlsanity.com/paper/2607.04525/) (2026)
- [A space of goals: the cognitive geometry of informationally bounded agents](https://mlsanity.com/paper/2111.03699/) (2021)

## Abstract (arXiv)

Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial neural networks (ANNs). While these two fields have different tools and applications, they present a similar challenge: namely, understanding how information is embedded and processed through high-dimensional representations to solve complex tasks. One approach to addressing this challenge is to utilize mathematical and computational tools to analyze the geometry of these high-dimensional representations, i.e., neural population geometry. We review examples of geometrical approaches providing insight into the function of biological and artificial neural networks: representation untangling in perception, a geometric theory of classification capacity, disentanglement and abstraction in cognitive systems, topological representations underlying cognitive maps, dynamic untangling in motor systems, and a dynamical approach to cognition. Together, these findings illustrate an exciting trend at the intersection of machine learning, neuroscience, and geometry, in which neural population geometry provides a useful population-level mechanistic descriptor underlying task implementation. Importantly, geometric descriptions are applicable across sensory modalities, brain regions, network architectures and timescales. Thus, neural population geometry has the potential to unify our understanding of structure and function in biological and artificial neural networks, bridging the gap between single neurons, populations and behavior.