# Euclidean equivariant neural network

I will explain this slide:

# Group

$$g_1, g_2 \in G$$

$$g_1 \circ g_2$$

neutral element

inverse

# Irreducible representations

A kind of data that transforms in a certain way under the group action

$$\xrightarrow{g}$$

$$\xrightarrow{g}$$

vector space on which the group acts linearly

irreducible representations of $$G$$

Direct sum

Tensor product

$$\oplus$$

$$=$$

$$\otimes$$

$$=$$

bigger vector space $$d=d_1+d_2$$

bigger vector space $$d=d_1 d_2$$

irreducible representations of $$G$$

$$\otimes$$

$$=$$

$$\oplus$$

$$=$$

one number

three numbers

five numbers

seven numbers

nine numbers

eleven numbers

...

one number

three numbers

five numbers

seven numbers

nine numbers

eleven numbers

...

# Irreps of $$O(3)$$

odd

even

$$d=1$$

($$L=0$$)

$$d=3$$

($$L=1$$)

$$d=5$$

($$L=2$$)

# Irreps of $$O(3)$$

odd

even

$$d=1$$

($$L=0$$)

$$d=3$$

($$L=1$$)

$$d=5$$

($$L=2$$)

$$d=7$$

($$L=3$$)

$$d=9$$

($$L=5$$)

$$d=11$$

($$L=6$$)

# Tensor Product

$$D_{L} \otimes D_{J} = D_{|L-J|} \oplus \dots \oplus D_{L+J}$$

## Example

$$1_o \otimes 2_e = 1_o \oplus 2_o \oplus 3_o$$

$$\otimes$$             =           $$\oplus$$               $$\oplus$$

parity simply multiply

$$\otimes$$             =           $$\oplus$$               $$\oplus$$

## Examples

$$\otimes$$             =                     $$\oplus$$          $$\oplus$$

$$1_e \otimes 1_o = 0_o \oplus 1_o \oplus 2_o$$

$$2_e \otimes 3_o = 1_o \oplus 2_o \oplus 3_o \oplus 4_o \oplus 5_o$$

$$\otimes$$           =           $$\oplus$$         $$\oplus$$            $$\oplus$$          $$\oplus$$

# Recap

• Addition $$x+y$$ from the save vector space to the same vector space
• Direct Sum $$x\oplus y$$ accolate together in a bigger vector space
• Tensor Product $$x \otimes y$$ multiply together in a bigger vector space

### All these operations are equivariant

• $$gx + gy = g(x+y)$$
• $$gx \oplus gy = g(x \oplus y)$$
• $$gx \otimes gy = g(x \otimes y)$$

# Paths in $$\otimes$$

### $$x$$

$$0_e$$

$$0_o$$

$$1_e$$

$$1_o$$

$$2_e$$

$$2_o$$

$$3_e$$

$$3_o$$

...

### $$y$$

$$0_e$$

$$0_o$$

$$1_e$$

$$1_o$$

$$2_e$$

$$2_o$$

$$3_e$$

$$3_o$$

...

### $$x\otimes y$$

$$0_e$$

$$0_o$$

$$1_e$$

$$1_o$$

$$2_e$$

$$2_o$$

$$3_e$$

$$3_o$$

...

# Paths in $$\otimes$$

### $$x$$

$$0_e$$

$$0_o$$

$$1_e$$

$$1_o$$

$$2_e$$

$$2_o$$

$$3_e$$

$$3_o$$

...

### $$y$$

$$0_e$$

$$0_o$$

$$1_e$$

$$1_o$$

$$2_e$$

$$2_o$$

$$3_e$$

$$3_o$$

...

### $$x\otimes y$$

$$0_e$$

$$0_o$$

$$1_e$$

$$1_o$$

$$2_e$$

$$2_o$$

$$3_e$$

$$3_o$$

...

## The Convolution operation...

• it interacts features $$x$$ with geometry $$\vec r$$
• is equivariant
• is linear in $$x$$ (for a start)
• is of the form $$y_a = \sum_b L(\vec r_{ab}, x_b)$$ (for a start)

linear & equivariant  $$\Rightarrow L(\vec r, x) = H(\vec r) \otimes x$$

$$H$$ is any equivariant function that outputs all the needed representations

$$H(g \vec r) = g H(\vec r)$$

$$Y$$ are the spherical harmonics

$$R$$ is any scalar function

$$\Rightarrow H(\vec r) = R(r) Y(\frac{\vec r}{r})$$

$$H(\vec r) = H(g_{\vec r} \vec e_z) = g_{\vec r} H(\vec e_z)$$

$$H(\vec e_z) = H(g_z \vec e_z) = g_z H(\vec e_z)$$

$$\Leftrightarrow$$ spherical harmonics

Spherical harmonics

$$Y^L(g\vec r) = g Y^L(\vec r)$$

0

+1

-1

$$Y^0$$

$$Y^1$$

$$Y^2$$

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

$$\displaystyle y_a = \sum_{b, \text{path}} R(\|\vec r_{ab}\|, \omega) \; Y^L(\frac{\vec r_{ab}}{\|\vec r_{ab}\|}) \otimes x_b$$

learned function

spherical harmonics

tensor product

linear

equivariant

1-neighbor

one degrees of freedom per path

By Mario Geiger

• 652