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