Geometric Deep Learning: One Blueprint Behind Every Architecture
Published:
The blueprint
Bronstein, Bruna, Cohen and Veličković’s Geometric Deep Learning programme makes a claim that sounds too neat to be useful and turns out to be extremely useful: most successful deep learning architectures are derivable from a single design procedure.
- Name the domain. A grid, a set, a graph, a manifold, a group.
- Name its symmetry group — the transformations under which the labels do not change. Shifting an image does not change what is in it. Relabelling the nodes of a molecule does not change the molecule.
- Build layers that are equivariant to that group. Transform the input, and the output transforms the same way.
- Add a pooling or readout step that is invariant, so the final prediction does not depend on the arbitrary choice at all.
The same recipe, five domains
| Domain | Symmetry | What the recipe produces | Where on this site |
|---|---|---|---|
| Grid | translation | convolution | CNNs |
| Set | permutation | attention / sum-pooling | Transformers |
| Graph | permutation, respecting edges | message passing | GNNs |
| Manifold | isometry | geodesic convolution | — |
| Local frames | gauge transformation | sheaf / connection Laplacians | Sheaf models |
Reading down that table is the argument of this book. A convolution is not an image-processing trick that happened to work; it is what you get when you demand translation equivariance on a grid. Message passing is not a graph heuristic; it is what you get when you demand permutation equivariance on a graph. And the sheaf machinery in Book III is what you get when you stop assuming all nodes share one coordinate frame — the gauge row.
The claim this book is built around
The sharpest instance of the blueprint is an equivalence rather than an analogy:
A Transformer is a graph neural network running message passing on a fully connected graph of tokens.
Not “is like”. The update equations are the same equations. Self-attention is message passing where the neighbourhood \(\mathcal{N}_i\) is the whole input, the message function is attention-weighted, and the aggregator is a sum. A Graph Attention Network is the same thing with the neighbourhood restricted by the adjacency matrix.
Chaitanya Joshi’s Transformers are Graph Neural Networks sets this out carefully, and the chapters here follow its structure:
- Permutation Symmetry and the Message-Passing Blueprint — what permutation equivariance actually requires, and why the three-step message/aggregate/update template is forced rather than chosen.
- Transformers Are GNNs on Fully Connected Graphs — the equivalence line by line, plus the two places where the identification is looser than usually stated.
- The Hardware Lottery — why, given the equivalence, the dense version won anyway. This is the most practically consequential chapter and the least mathematical.
Why this ordering of the site makes sense
Each of the other books teaches an architecture as if it were its own thing, which is the right way to learn it the first time. This book is the second pass, where the separate things turn out to be one thing:
- Book 0’s convolution is the grid row. Its fixed local receptive field is a choice of symmetry group, and that choice is what attention later relaxes.
- Book I’s Transformer is the set row, and — per the equivalence above — also the fully-connected-graph row.
- Book II’s GNN is the graph row. Its
oversquashingandoversmoothingpathologies are consequences of committing to a sparse graph, which the set view simply does not have. - Book III’s sheaf is the gauge row: what happens when you drop the assumption that every node’s features live in a shared coordinate system.
✅ Key Takeaways
- The blueprint is: name the domain, name its symmetry group, build equivariant layers, pool invariantly.
- Equivariant layers, invariant readout. \(f(g \cdot x) = g \cdot f(x)\) against \(f(g \cdot x) = f(x)\) — the distinction is load-bearing throughout.
- Convolution, attention, message passing and sheaf diffusion are four outputs of the same procedure applied to grids, sets, graphs and local frames.
- Transformers and GNNs are not analogous — a Transformer is message passing on a complete graph, with the same update equations.
- The blueprint constrains what is admissible. It says nothing about what trains well or runs fast, and on that second question the theory and the hardware disagreed.
References
- Bronstein, M. M., Bruna, J., Cohen, T., & Veličković, P. (2021). Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges. arXiv:2104.13478.
- Joshi, C. K. (2025). Transformers are Graph Neural Networks. arXiv:2506.22084.
- Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., & Bengio, Y. (2018). Graph Attention Networks. ICLR 2018.
- Battaglia, P. W., Hamrick, J. B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V., et al. (2018). Relational Inductive Biases, Deep Learning, and Graph Networks. arXiv:1806.01261.