$\longrightarrow \quad U_0$

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 $G$

irreducible representations of $G$

$\otimes$

$=$

$\longrightarrow \quad U_0$

Caffeine

What I want to do

input data

$F_a$

$F_b$

Equivariance

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

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

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

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

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

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

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

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

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

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

Can we formalize this a bit?

irreducible representations of SO(3)

$D^L(g)$

for $L=0, 1, 2, 3, \dots$

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

Spherical harmonics

0

+1

-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