# $$\longrightarrow \quad U_0$$

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

By Mario Geiger

• 648