Convolutional neural nets going to spacetime 🚀
Our new ICML24 paper shows how to build Lorentz-equivariant CNNs/MPNNs for multivector fields on Minkowski spaces. This is useful for particle physics or Navier Stokes / electrodynamics simulations.

🧵1/N arxiv.org/abs/2402.14730
Equidistant points in Euclidean space ℝ^N lie on spherical shells. For Minkowski spacetime ℝ^{1,N} they lie on hyperboloids since time is measured negatively. Distance preserving lin. symm. are rotations/reflections O(N) for ℝ^N and Lorentz trafos O(1,N) for ℝ^{1,N}.
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This is the 3rd post in our series on equivariant NNs. It gives an intuition for the representation theory of equivariant CNNs on Euclidean spaces ℝᵈ. These models rely necessarily on symmetry-constrained "G-steerable" kernels/biases/etc.

👇TL;DR maurice-weiler.gitlab.io/blog_post/cnn-…
Instead of restricting to one specific group, we allow for *any* affine groups Aff(G). They always include translations (ℝᵈ,+). In addition, they contain any choice of matrix (sub)group G≤GL(d), which allows to model rotations/reflections/dilations/shearing/...
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We proudly present our 524 page book on equivariant convolutional networks.
Coauthored by Patrick Forré, @erikverlinde and @wellingmax.

[1/N] maurice-weiler.gitlab.io/#cnn_book
The book brings together our findings on the representation theory and differential geometry of equivariant CNNs that we have obtained in recent years. It generalizes previous results, presents novel insights and adds background knowledge/intuition/visualizations/examples.
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Happy to announce our work on Coordinate Independent Convolutional Networks.
It develops a theory of CNNs on Riemannian manifolds and clarifies the interplay of the kernels' local gauge equivariance and the networks' global isometry equivariance.
arxiv.org/abs/2106.06020
[1/N] Why *coordinate independent* networks? In contrast to Euclidean spaces R^d, general manifolds do not come with a canonical choice of reference frames. This implies in particular that the alignment of a shared convolution kernel is inherently ambiguous.
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How to parameterize group equivariant CNNs?
Our generalization of the famous Wigner-Eckart theorem from quantum mechanics to G-steerable (equivariant) convolution kernels answers this question in a quite general setting.
Joint work with @Lang__Leon
arxiv.org/pdf/2010.10952…
[1/n] Recent work by @TacoCohen and @wellingmax proved that *any* equivariant convolution requires G-steerable kernels, satisfying a linear constraint depending on the features' group representations rho_in and rho_out.
arxiv.org/pdf/1811.02017…
arxiv.org/pdf/1807.02547…
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