Symmetry and Geometry in Neural Representations

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
  • Differential Geometry + Lie Groups
  • Algebra
  • Topology
  • Geometric Machine Learning
  • Computational Neuroscience
• Datasets
• Software Libraries
• Conferences and Workshops
• Research Papers
  • Perspectives
  • Neuroscience
    • Vision
    • Motor Control
    • Spatial Navigation
    • Abstract Representations
    • Methods
  • Geometric Machine Learning
    • Theory
    • Estimation on Manifolds
    • Dimensionality Reduction and Disentangling
    • Group-Invariant and -Equivariant Representation Learning

Educational Resources

Differential Geometry + Lie Groups

Textbooks

• Lie Groups, Lie Algebras, and Representations (2003)
  Brian C. Hall
• Differential Geometry and Lie Groups: A Computational Perspective (2020)
  Gallier & Quaintance

Courses, Lectures, and Videos

• 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

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

Algebra

Textbooks

• Tensors in Computations (2021)
  Lek-Heng Lim
• Aspects of Harmonic Analysis and Representation Theory (2021)
  Gallier & Quaintance
• Representation Theory of Finite Groups (2012)
  Bemjamin Steinberg

Courses, Lectures, and Videos

• What is a Tensor?
  XylyXylyX
• Representation Theory
  Math Doctor Bob

Topology

Courses, Lectures, and Videos

• Computational Algebraic Topology
  Vidit Nanda

Geometric Machine Learning

Textbooks

• Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021)
  Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković
• Group Theoretical Methods in Machine Learning (2008)
  Risi Kondor, PhD Thesis
• 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

• 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

• CSC 2547: Current Topics in Machine Learning Methods in 3D and Geometric Deep Learning (2021)
  Animesh Garg @ University of Toronto

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

• Rythyms 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

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 harmoics
• 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

Conferences and Workshops

• 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

Perspectives

• Symmetry-based representations for artificial and biological general intelligence (2022)
  Irina Higgins, Sébastien Racanière, Danilo Rezende.
  

Neuroscience

Vision

• 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
  

• Parsimony, Exhaustivity and Balanced Detection in Neocortex (2015)
  Alberto Romagnoni, Jérôme Ribot, Daniel Bennequin, Jonathan Touboul

• 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

Motor Control

• Neural manifolds for the control of movement (2017)
  Juan Gallego, Matthew Perich, Lee Miller, Sara Solla

Spatial Navigation

• 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
  

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

• 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

Geometric Machine Learning

Theory

• The Riemannian Geometry of Deep Generative Models (2018)
  Hang Shao, Abhishek Kumar, P. Thomas Fletcher
  

• 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
  

Group-Invariant and -Equivariant 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
  

• Group equivariant convolutional networks (2016)
  Taco Cohen & Max Welling
  

• 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