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Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation.
While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications.
Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.
While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications.
Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.
160 pages, ebook
Published May 4, 2021
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Displaying 1 - 3 of 3 reviews
August 17, 2022
A fantastic book that tries to obtain systematization of the field of Deep Learning.
The knowledge in this book will allow you to derive popular deep learning architectures from first principles. And because machine learning is such a fast-paced field where new architectures come out on a daily basis, that knowledge can help you make better connections between new model architectures and the old ones, and that will result in a faster and easier learning process.
The knowledge in this book will allow you to derive popular deep learning architectures from first principles. And because machine learning is such a fast-paced field where new architectures come out on a daily basis, that knowledge can help you make better connections between new model architectures and the old ones, and that will result in a faster and easier learning process.
April 22, 2023
By far my favorite ML-related paper (book?) to come out of the past 5 years. Actually lays out a principled way to design networks, and it does it all with geometric tools. Would especially recommend to those excited by mathematical physics
April 29, 2026
Geometric Deep Learning
I read the available book draft, the lecture notes, the paper, and watched the lectures.
The book is not finished yet. Some parts are much more polished than others. Some chapters go deep. Some topics are only sketched. So this is not a normal “finished textbook” experience.
But the core idea is just amazing.
The book tries to explain neural network architectures from geometry.
Not in the vague “geometry is everywhere” sense.
More concretely:
If your data lives on a grid, translations matter -> This leads to CNNs.
If your data is a set or a graph, the order of nodes should not matter -> This leads to Deep Sets, GNNs, and Transformers.
If your data lives on a sphere, rotations matter -> This leads to spherical CNNs.
If your data lives on a manifold, local geometry matters -> Now you need geodesics, tangent spaces, local frames, gauges, and bundles.
The main message is simple:
Architecture is a way to put assumptions into the model.
If you know the structure of the problem, you should not make the network rediscover it from scratch.
This is the same attitude that works so well in physics.
In physics, symmetries are not decorative. They restrict what the laws can be.
- Time translation gives energy conservation.
- Space translation gives momentum conservation.
- Gauge symmetry gives electric charge conservation and electromagnetism in general.
Geometric Deep Learning applies a similar idea to ML.
Instead of asking:
Which architecture should I try?
you ask:
- What is the domain?
- What transformations should preserve the answer?
- What transformations should move the answer in a predictable way?
- What locality structure exists?
- What should be invariant?
- What should be equivariant?
This reframes a lot of modern deep learning.
CNNs, GNNs, Transformers, Deep Sets, equivariant networks, mesh CNNs, spherical CNNs — they stop looking like separate tricks. They become different points in the same design space.
This was the most useful part of the book for me.
The book is also hard.
You need some comfort with:
- groups,
- representations,
- graphs,
- manifolds,
- differential geometry,
- Lie groups,
- bundles,
- and modern ML.
The gauges and bundles part is probably the hardest one. It is also one of the most interesting parts, because it connects directly to physics: local frames, gauge freedom, connections, and curvature.
But I would not recommend starting here if you do not already know basic ML architectures.
This is not a first ML book.
This is a book for the moment when you already know what CNNs, GNNs, and Transformers are, but you want to understand why these things have the shapes they have.
The strongest idea I took from the book:
A neural network is not just a stack of layers. It is a mathematical object with built-in assumptions about the data.
When those assumptions match the task, the model trains faster, generalizes better, and needs less data.
When they do not match, you just hard-code the wrong bias.
So the practical question becomes: "What geometry does my problem actually have?"
That question is much more useful than blindly stacking popular architectures.
Overall: unfinished, uneven, demanding, but very valuable.
I would recommend it to people interested in the theory of ML, especially if they like the intersection of deep learning, geometry, graphs, physics, and symmetry.
Not recommended as a casual read.
Very recommended if you want to understand why architecture design is not just engineering, but also geometry.
Rating: 5/5 for the ideas. 4/5 as a book in its current unfinished form.
I read the available book draft, the lecture notes, the paper, and watched the lectures.
The book is not finished yet. Some parts are much more polished than others. Some chapters go deep. Some topics are only sketched. So this is not a normal “finished textbook” experience.
But the core idea is just amazing.
The book tries to explain neural network architectures from geometry.
Not in the vague “geometry is everywhere” sense.
More concretely:
If your data lives on a grid, translations matter -> This leads to CNNs.
If your data is a set or a graph, the order of nodes should not matter -> This leads to Deep Sets, GNNs, and Transformers.
If your data lives on a sphere, rotations matter -> This leads to spherical CNNs.
If your data lives on a manifold, local geometry matters -> Now you need geodesics, tangent spaces, local frames, gauges, and bundles.
The main message is simple:
Architecture is a way to put assumptions into the model.
If you know the structure of the problem, you should not make the network rediscover it from scratch.
This is the same attitude that works so well in physics.
In physics, symmetries are not decorative. They restrict what the laws can be.
- Time translation gives energy conservation.
- Space translation gives momentum conservation.
- Gauge symmetry gives electric charge conservation and electromagnetism in general.
Geometric Deep Learning applies a similar idea to ML.
Instead of asking:
Which architecture should I try?
you ask:
- What is the domain?
- What transformations should preserve the answer?
- What transformations should move the answer in a predictable way?
- What locality structure exists?
- What should be invariant?
- What should be equivariant?
This reframes a lot of modern deep learning.
CNNs, GNNs, Transformers, Deep Sets, equivariant networks, mesh CNNs, spherical CNNs — they stop looking like separate tricks. They become different points in the same design space.
This was the most useful part of the book for me.
The book is also hard.
You need some comfort with:
- groups,
- representations,
- graphs,
- manifolds,
- differential geometry,
- Lie groups,
- bundles,
- and modern ML.
The gauges and bundles part is probably the hardest one. It is also one of the most interesting parts, because it connects directly to physics: local frames, gauge freedom, connections, and curvature.
But I would not recommend starting here if you do not already know basic ML architectures.
This is not a first ML book.
This is a book for the moment when you already know what CNNs, GNNs, and Transformers are, but you want to understand why these things have the shapes they have.
The strongest idea I took from the book:
A neural network is not just a stack of layers. It is a mathematical object with built-in assumptions about the data.
When those assumptions match the task, the model trains faster, generalizes better, and needs less data.
When they do not match, you just hard-code the wrong bias.
So the practical question becomes: "What geometry does my problem actually have?"
That question is much more useful than blindly stacking popular architectures.
Overall: unfinished, uneven, demanding, but very valuable.
I would recommend it to people interested in the theory of ML, especially if they like the intersection of deep learning, geometry, graphs, physics, and symmetry.
Not recommended as a casual read.
Very recommended if you want to understand why architecture design is not just engineering, but also geometry.
Rating: 5/5 for the ideas. 4/5 as a book in its current unfinished form.
Displaying 1 - 3 of 3 reviews