Awesome Neural Geometry

A curated collection of resources and research related to the geometry of representations in the brain, deep networks, and beyond, collaboratively generated on the Symmetry and Geometry in Neural Representations Slack Workspace.

This is a collaborative work-in-progress. Please contribute via PRs!

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Contents

• Educational Resources
  • Abstract Algebra + Group Theory
  • Differential Geometry + Lie Groups
  • Information Geometry
  • Topology
  • Geometric Machine Learning
  • Computational Neuroscience
• Datasets
• Software Libraries
• Conferences and Workshops
• Research Papers
  • Neuroscience
    • Theory and Perspectives
    • Vision
    • Motor Control
    • Spatial Navigation
    • Abstract Representations
    • Methods
  • Geometric Machine Learning
    • Theory
    • Estimation on Manifolds
    • Dimensionality Reduction and Disentangling
    • Deep Network Interpretability
    • Group-Invariant Representation Learning
    • Generalized Convolutional Neural Networks

Educational Resources

Abstract Algebra + Group Theory

Textbooks & Notes

• Group Theory: A Primer (2019)
  Luciano da Fontoura Costa
• Tensors in Computations (2021)
  Lek-Heng Lim
• Aspects of Harmonic Analysis and Representation Theory (2021)
  Gallier & Quaintance
• Basic concepts of representation theory (2013)
  Amritanshu Prasad
• Representation Theory of Finite Groups (2012)
  Bemjamin Steinberg

Courses, Lectures, and Videos

• Essence of Group Theory *Beginner-Friendly
  Mathemaniac
• Abstract Algebra *Beginner-Friendly
  Socratica
• Euler's formula with introductory group theory *Intuition Building
  3blue1brown
• What is a Tensor?
  XylyXylyX
• Representation Theory
  Math Doctor Bob
• Category Theory for AI
  Online Course, October 2022

Differential Geometry + Lie Groups

Textbooks & Notes

• Lie Groups, Lie Algebras, and Representations (2003)
  Brian C. Hall
• Differential Geometry and Lie Groups: A Computational Perspective (2020)
  Gallier & Quaintance
• Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats (2022)
  Nicolas Guigui, Nina Miolane, Xavier Pannec

Courses, Lectures, and Videos

• Geometry and Topology and Symmetry *Beginner-Friendly
  Sean Carroll
• Differential Geometry for Computer Science
  Justin Solomon
• Discrete Differential Geometry
  CMU
• What is a Manifold?
  XylyXylyX
• Manifolds
  Robert Davie
• Lie Groups and Lie Algebras
  XylyXylyX
• Lectures on Geometric Anatomy of Theoretical Physics
  Frederic Schuller
• Weekend with Bernie (Riemann)
  Søren Hauberg @ DTU
• Riemann and Gauss Meet Asimov: A Tutorial on Geometric Methods in Robot Learning, Optimization, and Control
  IROS 2022

Notebooks and Blogposts

• Introduction to Differential Geometry and Machine Learning
  Geomstats Jupyter notebooks
• Differential Geometry for Machine Learning
  Roger Grosse
• Manifolds: A Gentle Introduction
  Brian Keng

Information Geometry

Primers

• The Many Faces of Information Geometry (2022)
  Frank Nielsen

Topology

Courses, Lectures, and Videos

• Computational Algebraic Topology
  Vidit Nanda

• Topological Data Analysis for Machine Learning
  Bastian Rieck

Textbooks, Notes

• Elementary Applied Topology
  Robert Ghrist
  

Geometric Machine Learning

Textbooks

• Group Invariance Applications in Statistics (1989)
  Morris Eaton
• Group Theoretical Methods in Machine Learning (2008)
  Risi Kondor, PhD Thesis
• Pattern Theory: The Stochastic Analysis of Real-World Signals (2010)
  David Mumford and Agnès Desolneux
• Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021)
  Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković
• Equivariant Convolutional Networks (2021)
  Taco Cohen, PhD Thesis
• An Introduction to Optimization on Smooth Manifolds (2022)
  Nicolas Boumal

Courses, Lectures, and Videos

• Geometric Deep Learning (2nd Edition)
  Michael Bronstein, Joan Bruna, Taco Cohen, Petar Veličković @ AMMI
• CSC 2547: Current Topics in Machine Learning Methods in 3D and Geometric Deep Learning (2021)
  Animesh Garg @ University of Toronto
• An Introduction to Group-Equivariant Deep Learning (2022)
  Erik Bekkers @ UvA
• Italian Summer School on Geometric Deep Learning (2022)
  Cristian Bodnar, Michael Bronstein, Francesco Di Giovanni, Pim de Haan, Maurice Weiler
• COMP760: Geometry and Generative Models (2022)
  Joey Bose and Prakash Panangaden @ MILA

Notebooks and Blogposts

• Geometric foundations of Deep Learning
  Michael Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković
• What does 2022 hold for Geometric & Graph ML?
  Michael Bronstein
• Geometric Machine Learning for Shape Analysis with Jupyter Notebooks
  Nina Miolane

Computational Neuroscience

Textbooks

• Introduction to the Theory of Neural Computation (1991)
  John Hertz, Anders Krogh, Richard G Palmer
• Theoretical Neuroscience (2001)
  Peter Dayan
• Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (2006)
  Eugene M. Izhikevich
• Principles of Neural Design (2015)
  Peter Sterling & Simon Laughlin

Books - General Audience

• Rhythms of the Brain (2006)
  Gyorgy Buzsaki
• Networks of the Brain (2010)
  Olaf Sporns
• Models of the Mind: How Physics, Engineering and Mathematics Have Shaped Our Understanding of the Brain (2021)
  Grace Lindsay

Tutorials

• Generative Models for Neural Data Analysis
  COSYNE Workshops 2023

Courses, Lectures, and Videos

• Neural Computation VS265
  Bruno Olshausen @ UC Berkeley

Datasets

Open-Source Neuroscience Datasets

• OpenNeuro
• NeuroVault
• CRCNS
• NeuroData Without Borders
• Allen Brain Atlas
• Kavli Institute for Systems Neuroscience Grid Cell Database
• The Natural Scenes Dataset

Software Libraries

• Geomstats
  • Computation, statistics, and machine learning on non-Euclidean manifolds
• Giotto TDA
  • Topological Data Analysis
• E3NN
  • E(3)-equivariant neural networks
• equivariant-MLP
  • Construct equivariant multilayer perceptrons for arbitrary matrix groups
• SHTOOLS
  • Python library for computations involving spherical harmonics
• LieConv
  • Generalizing convolutional neural networks for equivariance to Lie groups on arbitrary continuous data
• Open Neuroscience
  • A database of open-source tools and software for neuroscience
• LieTorch
  • Geometric machine learning and Lie analysis tools for PyTorch

Conferences and Workshops

• NeurIPS Workshop on Symmetry and Geometry in Neural Representations (2023)
• ICML Workshop on Topology, Algebra and Geometry in Machine Learning (2023)
• RSS Workshop on Symmetries in Robot Learning (2023)
• NeurIPS Workshop on Symmetry and Geometry in Neural Representations (2022)
• ICML Workshop on Topology, Algebra and Geometry in Machine Learning (2022)
• ICLR Workshop on Geometrical and Topological Representation Learning (2022)
• Workshop on Symmetry, Invariance and Neural Representations @ The Bernstein Conference (2022)

Research Papers

Math Tags

Neuroscience

Theory and Perspectives

• Brain graphs: graphical models of the human brain connectome. (2011)
  Edward T. Bullmore and Danielle S. Bassett
  
• What can topology tell us about the neural code? (2017)
  Carina Curto
  
• Network Neuroscience (2017)
  Danielle S. Bassett, and Olaf Sporns.
  
• Navigating the neural space in search of the neural code. (2017)
  Jazayeri, M., & Afraz, A.
• A theory of multineuronal dimensionality, dynamics and measurement. (2017)
  Gao, P., Trautmann, E., Yu, B., Santhanam, G., Ryu, S., Shenoy, K., & Ganguli, S.
• Computation through neural population dynamics. (2020)
  Saurabh Vyas, Matthew D. Golub, David Sussillo, and Krishna V. Shenoy
• Neural population geometry: An approach for understanding biological and artificial neural networks. (2021)
  SueYeon Chung, and L. F. Abbott.
• Symmetry-based representations for artificial and biological general intelligence (2022)
  Irina Higgins, Sébastien Racanière, Danilo Rezende.
  

Vision

• Mathematical analysis of binocular vision (1947)
  Rudolf Karl Luneburg
  
• The Lie algebra of visual perception (1966)
  William C.Hoffman
  
• Representation of local geometry in the visual system (1987)
  Jan Koenderink
• Operational Significance of Receptive Fields Assemblies (1989)
  Jan Koenderink
• The Visual Cortex is a Contact Bundle (1989)
  William C. Hoffman
  
• Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex (2001)
  Paul C. Bressloff, Jack D. Cowan, Martin Golubitsky, Peter J. Thomas and Matthew C. Wiener
• The neurogeometry of pinwheels as a sub-Riemannian contact structure (2003)
  Jean Petitot
  
• Untangling invariant object recognition. (2007)
  James DiCarlo and David Cox
• Parsimony, Exhaustivity and Balanced Detection in Neocortex (2015)
  Alberto Romagnoni, Jérôme Ribot, Daniel Bennequin, Jonathan Touboul
• High-dimensional geometry of population responses in visual cortex. (2019)
  Stringer, Carsen, Marius Pachitariu, Nicholas Steinmetz, Matteo Carandini, and Kenneth D. Harris.
• Evolving images for visual neurons using a deep generative network reveals coding principles and neuronal preferences. (2019)
  Carlos R. Ponce, Will Xiao, Peter F. Schade, Till S. Hartmann, Gabriel Kreiman, and Margaret S. Livingstone.
• Predicting the retinotopic organization of human visual cortex from anatomy using geometric deep learning (2021)
  Fernanda L. Ribeiro, Steffen Bollmann, Alexander M. Puckett
• Primary visual cortex straightens natural video trajectories (2021)
  Olivier J. Hénaff, Yoon Bai, Julie A. Charlton, Ian Nauhaus, Eero P. Simoncelli & Robbe L. T. Goris
• Unsupervised deep learning identifies semantic disentanglement in single inferotemporal face patch neurons. (2021)
  Higgins, I., Chang, L., Langston, V., Hassabis, D., Summerfield, C., Tsao, D., & Botvinick, M.

Motor Control

• Neural manifolds for the control of movement (2017)
  Juan Gallego, Matthew Perich, Lee Miller, Sara Solla
• Accurate estimation of neural population dynamics without spike sorting. (2019)
  Eric M. Trautmann, Sergey D. Stavisky, Subhaneil Lahiri, Katherine C. Ames, Matthew T. Kaufman, Daniel J. O’Shea, Saurabh Vyas et al.

Spatial Navigation

• Ring attractor dynamics in the Drosophila central brain (2017)
  Sung Soo Kim, Hervé Rouault, Shaul Druckmann, and Vivek Jayaraman
• The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit across waking and sleep (2019)
  Rishidev Chaudhuri, Berk Gerçek, Biraj Pandey, Adrien Peyrache & Ila Fiete
  
• Toroidal topology of population activity in grid cells (2022)
  Richard J. Gardner, Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A. Baas, Benjamin A. Dunn, May-Britt Moser & Edvard I. Moser
  
• Hippocampal spatial representations exhibit a hyperbolic geometry that expands with experience (2022)
  Huanqiu Zhang, P. Dylan Rich, Albert K. Lee & Tatyana O. Sharpee
  

Abstract Representations

• The Geometry of Abstraction in the Hippocampus and Prefrontal Cortex (2020)
  Silvia Bernardi, Marcus K.Benna, Matti Rigotti, Jérôme Munuera, Stefano Fusi, Daniel Salzman
• Geometry of abstract learned knowledge in the hippocampus (2021)
  Edward Nieh, et al.

Methods

• Clique topology reveals intrinsic geometric structure in neural correlations. (2015)
  Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov
  
• Two’s company, three (or more) is a simplex. (2016)
  Chad Giusti, Robert Ghrist, and Danielle S. Bassett.
  
• Inferring single-trial neural population dynamics using sequential auto-encoders. (2018)
  Pandarinath, C., O’Shea, D. J., Collins, J., Jozefowicz, R., Stavisky, S. D., Kao, J. C., ... & Sussillo, D.
• Manifold GPLVMs for discovering non-Euclidean latent structure in neural data (2020)
  Kristopher Jensen, Ta-Chu Kao, Marco Tripodi, and Guillaume Hennequin
  
• Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity (2021)
  Mehrdad Jazayeri, Srdjan Ostojic
• Generalized shape metrics on neural representations. (2021)
  Williams, A. H., Kunz, E., Kornblith, S., & Linderman, S.
  
• Learnable latent embeddings for joint behavioural and neural analysis (2023)
  Schneider, S., Lee, J.H., Mathis, M.W.
  

Geometric Machine Learning

Theory

• On the Local Behavior of Spaces of Natural Images (2006)
  Gunnar Carlsson, Tigran Ishkhanov, Vin De Silva, and Afra Zomorodian
  
• The Riemannian Geometry of Deep Generative Models (2018)
  Hang Shao, Abhishek Kumar, P. Thomas Fletcher
  
• The Geometry of Deep Generative Image Models and its Applications (2021)
  Binxu Wang, Carlos R. Ponce
  
• Universal Approximation Theorems for Differentiable Geometric Deep Learning (2022)
  Anastasis Kratsios, L´eonie Papon
  
• A Statistical Manifold Framework for Point Cloud Data (2022)
  Yonghyeon Lee, Seungyeon Kim, Jinwon Choi, Frank C. Park
  

Estimation on Manifolds

• Intrinsic Statistics on Riemannian Manifolds (2011)
  Xavier Pennec
  
• Geometric Statistics for Computational Anatomy (2016)
  Nina Miolane, PhD Thesis
  
• Maximum Likelihood Estimators on Manifolds (2017)
  Hatem Hajri, Salem Said, Yannick Berthoumieu
  

Dimensionality Reduction and Disentangling

• Hyperspherical Variational Auto-Encoders (2018)
  Tim R. Davidson, Luca Falorsi, Nicola De Cao, Thomas Kipf, Jakub M. Tomczak
  
• Addressing the Topological Defects of Disentanglement via Distributed Operators (2021)
  Diane Bouchacourt, Mark Ibrahim, Stéphane Deny
  
• Visualizing Riemannian Data with Rie-SNE (2022)
  Andri Bergsson and Soren Hauberg
  
• Regularized Autoencoders for Isometric Representation Learning (2022)
  Yonghyeon Lee, Sangwoong Yoon, Minjun Son, Frank C. Park
  

Deep Network Interpretability

• Separability and geometry of object manifolds in deep neural networks. (2020)
  Uri Cohen, SueYeon Chung, Daniel D. Lee, and Haim Sompolinsky.
• Naturally occurring equivariance in neural networks. (2020)
  Chris Olah, Nick Cammarata, Chelsea Voss, Ludwig Schubert, and Gabriel Goh.
• The Geometry of Deep Generative Image Models and its Applications (2021)
  Binxu Wang, Carlos R. Ponce
  

Group-Invariant Representation Learning

• How we know universals (1947)
  Walter Pitts & Warren S. McCulloch
  
• Learning Symmetry Groups with Hidden Units: Beyond the Perceptron (1986)
  Terrence Sejnowski, Paul K. Kienker, Geoffrey Hinton
  
• Learning Lie groups for invariant visual perception (1999)
  Rajesh Rao, Daniel Ruderman
  
• Learning the Lie groups of visual invariance (2007)
  Xu Miao, Rajesh Rao
  
• An unsupervised algorithm for learning Lie group transformations. (2010)
  Jascha Sohl-Dickstein, Ching Ming Wang, Bruno Olshausen
  
• Learning the irreducible representations of commutative Lie groups (2014)
  Taco Cohen & Max Welling
  
• Bispectral Neural Networks (2022)
  Sophia Sanborn, Christian Shewmake, Bruno Olshausen, Christopher Hillar
  
• Geometric Clifford Algebra Networks (2023)
  David Ruhem Jayesh K. Gupta, Steven de Keninck, Max Welling, Johannes Brandstetter
  

Generalized Convolutional Neural Networks

Groups and Homogeneous Spaces

• Group equivariant convolutional networks (2016)
  Taco Cohen & Max Welling
  
• Steerable CNNs (2016)
  Taco Cohen & Max Welling
  
• Harmonic Networks (2017)
  Daniel E. Worrall, Stephan J. Garbin, Daniyar Turmukhambetov, and Gabriel J. Brostow.
  
• Spherical CNNs (2018)
  Taco Cohen, Mario Geiger, Jonas Kohler, & Max Welling
  
• Clebsch–Gordan nets: a fully Fourier space spherical convolutional neural network (2018)
  Risi Kondor, Zhen Lin, Shubhendu Trivedi
  
• Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds (2018)
  Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, Patrick Riley
  
• A General Theory of Equivariant CNNs on Homogeneous Spaces (2019)
  Taco Cohen, Mario Geiger, & Maurice Weiler
  
• Generalizing convolutional neural networks for equivariance to Lie groups on arbitrary continuous data (2020)
  Marc Finzi, Samuel Stanton, Pavel Izmailov, Andrew Gordon Wilson
  
• PDE-Based Group Equivariant Convolutional Neural Networks (2022)
  Bart M. N. Smets, Jim Portegies, Erik J. Bekkers & Remco Duits
  
• Clifford Neural Layers for PDE Modeling (2022)
  Johannes Brandstetter, Rianne van den Berg, Max Welling, Jayesh K. Gupta
  
• Clifford Group Equivariant Neural Networks (2023)
  David Ruhe, Johannes Brandstetter, Patrick Forré
  

Geodesics and Riemannian Manifolds

• Geodesic convolutional neural networks on Riemannian manifolds (2015)
  Jonathan Masci, Davide Boscaini, Michael Bronstein, and Pierre Vandergheynst
  
• Geometric deep learning on graphs and manifolds using mixture model CNNs (2017)
  Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M. Bronstein
  

Gauges and Bundles

• Gauge equivariant convolutional networks and the icosahedral CNN (2019)
  Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling
  
• Coordinate Independent Convolutional Networks - Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds (2021)
  Maurice Weiler, Patrick Forré, Erik Verlinde, Max Welling