Euclidean equivariant neural network

\(\longrightarrow \quad U_0\)

Caffeine

What I want to do

Euclidean group

SE(3) = rotation (SO(3)) + translations

\((\circ, G)\) is a group

  • \(e \in G\)
  • \(g_1 \circ g_2 \in G\)
  • \(\exists g^{-1}, g\circ g^{-1} = e\)

\((V, D)\) is a representation of the group \(G\)

  • \(D(e) = 1\)
  • \(D(g_1) D(g_2) = D(g_1 \circ g_2)\)
  • \(D(g)\) is a linear map \(V \to V\)
  • direct sum \(\oplus\)
  • tensor product \(\otimes\)
  • irreducible representations
  • reduction by change of basis \(A D(g) A^{-1} = D_1(g) \oplus \dots\)

irreducible representations of \(G\)

Direct sum

Tensor product

\(\oplus\)

\(=\)

\(\otimes\)

\(=\)

irreducible representations of \(G\)

\(  \otimes  \)

\(= \)

\(\longrightarrow \quad U_0\)

Caffeine

What I want to do

input data

\(F_a\)

\(F_b\)

Equivariance

 

\(f(D(R)x)=D'(R)f(x)\)

\(\vec F'_a\)

\(\vec x_{ba}\)

\(\vec F_b\)

\(\vec F'_a\)

\(\vec x_{ba}\)

\(\vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \| \vec x_{ba}\| \vec F_b\)

\(\vec F'_a\)

\(\vec x_{ba}\)

\(\vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \| \vec x_{ba}\| \vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \vec x_{ba} \wedge \vec F_b\)

\(\vec F'_a\)

\(\vec x_{ba}\)

\(\vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \vec x_{ba} (\vec x_{ba} \cdot \vec F_b) \)

\(\displaystyle \vec F'_a = \sum_b \| \vec x_{ba}\| \vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \vec x_{ba} \wedge \vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \| \vec x_{ba}\| \vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \vec x_{ba} \wedge \vec F_b\)

\(\displaystyle \vec F'_a = \sum_b \vec x_{ba} (\vec x_{ba} \cdot \vec F_b) \)

Can we formalize this a bit?

\(\vec F'_a\)

\(\vec x_{ba}\)

\(\vec F_b\)

irreducible representations of SO(3)

\(D^L(g)\)

for \(L=0, 1, 2, 3, \dots\)

\(\dim(D^L) = 2L+1\)

scalars (color, temperature, time left, ...)

vectors (velocity, gravity, dipole moment, ...)

traceless symmetric tensor (moment of inertia, stress tensor, ...)

\(3 \otimes 3 = 1 \oplus 3 \oplus 5 \)

\(D^{1} \otimes D^{1} = D^{0} \oplus D^{1} \oplus D^{2} \)

Spherical harmonics

\(Y^L(R\vec x) = D^L(R) Y^L(\vec x)\)

0

+1

-1

\(Y^0\)

\(Y^1\)

\(Y^2\)

\(Y^L: s^2 \longrightarrow \mathbb{R}^{2L+1}\)

\(\displaystyle F'_a = \sum_b R(\|\vec x_{ab}\|, \omega) \; Y^L(\frac{\vec x_{ab}}{\|\vec x_{ab}\|}) \otimes F_b\)

learned function

spherical harmonics

tensor product

linear

equivariant

1-neighbor

our method

Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties, K. Miller et al.

ongoing development

many possible architectures to explore

thanks :)

http://e3nn.org

e3nn-v1

By Mario Geiger

e3nn-v1

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