Equivariant Polynomials with e3nn-jax

Mario Geiger

Postdoc at

This Talk is about Equivariant Neural Networks

input

output

Illustration of a neural network equivariant to rotations in 3D

What is e3nn?

https://arxiv.org/pdf/2207.09453.pdf

https://github.com/e3nn/e3nn

What is e3nn?

What is e3nn?

Data efficient!

(Nequip: Simon Batzner et al. 2021)

max L of the messages

What is e3nn?

Data efficient!

(Nequip: Simon Batzner et al. 2021)

max L of the messages

With e3nn!

What is e3nn?

Data efficient!

(MACE: Ilyes Batatia et al. 2022)

L

L

3

\(m = (\sum_i h_i\otimes Y(\vec r_i))^{\otimes\nu}\)

(

)

What is e3nn?

Also a jax library 🚀

https://github.com/e3nn/e3nn-jax

Why Equivariant Polynomials?

All linear functions

Why Equivariant Polynomials?

All linear functions

All polynomials

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

Most physics is described by smooth functions

Random Facts about these classes of functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

Some Phase transitions are characterized by non continuous functions

Random Facts about these classes of functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

Some Phase transitions are characterized by non continuous functions

... that are limits of analytical functions (Landau theory)

Random Facts about these classes of functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

"Convergence of the training can then be related to the positive-definiteness of the
limiting NTK. We prove the positive-definiteness of the limiting NTK when the
data is supported on the sphere and the non-linearity is non-polynomial."

TNK Arthur Jacot 2020

Random Facts about these classes of functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

analytical functions are the limits of polynomials

Random Facts about these classes of functions

Why Equivariant Polynomials?

All linear functions

All polynomials

All analytical functions

All smooth functions

All continuous functions

All functions

It's a good start to be able to build any polynomial 🤷🏼‍♂️

Group and Representations

Group and Representations

"what are the operations"

"how they compose"

Group and Representations

"what are the operations"

"how they compose"

"vector spaces on which the action of the group is defined"

Group and Representations

"what are the operations"

"how they compose"

rotations, parity, (translations)

scalars, vectors, pseudovectors, ...

"vector spaces on which the action of the group is defined"

Group and Representations

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Equivalent notation \(D(g) x\)

  • \(D(g) : V\to V\)
  • \(D(g) \in \mathbb{R}^{d\times d}\)
  • \(D(gh) = D(g) D(h)\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Equivalent notation \(D(g) x\)

  • \(D(g) : V\to V\)
  • \(D(g) \in \mathbb{R}^{d\times d}\)
  • \(D(gh) = D(g) D(h)\)

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Equivalent notation \(D(g) x\)

  • \(D(g) : V\to V\)
  • \(D(g) \in \mathbb{R}^{d\times d}\)
  • \(D(gh) = D(g) D(h)\)

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Representation \(D(g, x)\)

  • \(g\in G\), \(x \in V\)
  • Linear \(D(g, x+y) = D(g,x) + D(g,y) \)
  • Follow the structure of the group
    \(D(gh,x) = D(g, D(h,x))\)

Group and Representations

"what are the operations" "how they compose"

"vector spaces on which the action of the group is defined"

Equivalent notation \(D(g) x\)

  • \(D(g) : V\to V\)
  • \(D(g) \in \mathbb{R}^{d\times d}\)
  • \(D(gh) = D(g) D(h)\)

Group \(G\)

  • \(\text{identity} \in G\)
  • associativity \(g (hk) = (gh)k\)
  • inverse \(g^{-1} \in G\)

Examples of representations

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

Examples of representations

3 scalars (3x0e)

Representations are like data types

It tells you how to interpret the data with respect to the group action

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

Examples of representations

3 scalars (3x0e)

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

Knowing that \(a_1, a_2, a_3\) are scalars tells you that they are not affected by a rotation of your system

Representations are like data types

It tells you how to interpret the data with respect to the group action

Examples of representations

3 scalars (3x0e)

a vector (1o)

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

Representations are like data types

It tells you how to interpret the data with respect to the group action

Examples of representations

3 scalars (3x0e)

a vector (1o)

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

If the system is rotated, the 3 components of the vector change together!

Representations are like data types

It tells you how to interpret the data with respect to the group action

Examples of representations

3 scalars (3x0e)

a vector (1o)

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

a vector (1o)

The two vectors transforms indepentently

Representations are like data types

It tells you how to interpret the data with respect to the group action

Examples of representations

3 scalars (3x0e)

a vector (1o)

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

a vector (1o)

system rotated by \(g\)

\(\begin{bmatrix} a'^1\\a'^2\\a'^3\\a'^4\\a'^5\\a'^6\\a'^7\\a'^8\\a'^9\end{bmatrix}=D(g)\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

Representations are like data types

It tells you how to interpret the data with respect to the group action

Examples of representations

3 scalars

a vector

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

a vector

system rotated by \(g\)

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

Representations are like data types

It tells you how to interpret the data with respect to the group action

\(\begin{bmatrix} 1&&&&&&&&&&\\&1&&&&&&&\\&&1&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\end{bmatrix}\)

\(\begin{bmatrix}&&\\&R\\&&\end{bmatrix}\)

\(\begin{bmatrix}&&\\&R\\&&\end{bmatrix}\)

\(\begin{bmatrix} a'^1\\a'^2\\a'^3\\a'^4\\a'^5\\a'^6\\a'^7\\a'^8\\a'^9\end{bmatrix}=\)

Examples of representations

3 scalars

a vector

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

a vector

system rotated by \(g\)

Representations are like data types

It tells you how to interpret the data with respect to the group action

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

\(\begin{bmatrix} 1&&&&&&&&&&\\&1&&&&&&&\\&&1&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\end{bmatrix}\)

\(\begin{bmatrix}&&\\&R\\&&\end{bmatrix}\)

\(\begin{bmatrix}&&\\&R\\&&\end{bmatrix}\)

\(\begin{bmatrix} a'^1\\a'^2\\a'^3\\a'^4\\a'^5\\a'^6\\a'^7\\a'^8\\a'^9\end{bmatrix}=\)

Examples of representations

3 scalars

a vector

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\in \mathbb{R}^9\)

a vector

system rotated by \(g\)

Representations are like data types

It tells you how to interpret the data with respect to the group action

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

\(\begin{bmatrix} 1&&&&&&&&&&\\&1&&&&&&&\\&&1&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\\&&&&&&&&\end{bmatrix}\)

\(\begin{bmatrix}&&\\&R\\&&\end{bmatrix}\)

\(\begin{bmatrix}&&\\&R\\&&\end{bmatrix}\)

\(\begin{bmatrix} a'^1\\a'^2\\a'^3\\a'^4\\a'^5\\a'^6\\a'^7\\a'^8\\a'^9\end{bmatrix}=\)

Equivariance

\(V\)

\(V'\)

Equivariance

\(V\)

\(V'\)

\(D(g)\)

\(D'(g)\)

\(V\)

\(V'\)

Equivariance

\(V\)

\(V'\)

\(D(g)\)

\(D'(g)\)

\(V\)

\(V'\)

\(f\)

Equivariance

\(V\)

\(V'\)

\(D(g)\)

\(D'(g)\)

\(V\)

\(V'\)

\(f\)

\(f\)

Equivariance

\(V\)

\(V'\)

\(D(g)\)

\(D'(g)\)

\(V\)

\(V'\)

\(f\)

\(f\)

\(f(D(g) x)\)

Equivariance

\(V\)

\(V'\)

\(D(g)\)

\(D'(g)\)

\(V\)

\(V'\)

\(f\)

\(f\)

\(f(D(g) x)\)

\(D'(g) f(x)\)

Equivariance

\(V\)

\(V'\)

\(D(g)\)

\(D'(g)\)

\(V\)

\(V'\)

\(f\)

\(f\)

\(f(D(g) x)\)

\(D'(g) f(x)\)

\(=\)

Polynomials

\(x \mapsto x^2 + 2(x-4)\)

Polynomials

\(x \mapsto x^2 + 2(x-4)\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto x^2 + 2(y-z)x\)

Polynomials

\(x \mapsto x^2 + 2(x-4)\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto x^2 + 2(y-z)x\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + 2(y-z) \\ z^4 + 100 x y z\end{bmatrix}\)

Equivariant Polynomials

\(P(D(g) x) = D'(g) P(x)\)

Equivariant Polynomials

\(P(D(g) x) = D'(g) P(x)\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + 2(y-z) \\ z^4 + 100 x y z \\ z\end{bmatrix}\)

Not equivariant

Equivariant Polynomials

\(P(D(g) x) = D'(g) P(x)\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + 2(y-z) \\ z^4 + 100 x y z \\ z\end{bmatrix}\)

Not equivariant

\(\begin{bmatrix} y\\ -x\\ z \end{bmatrix} \mapsto \begin{bmatrix} y^2 + 2(-x-z) \\ z^4 - 100 x y z\\ z\end{bmatrix}\)

\(R\)

not linear, probably not even invertible

\(?\)

Equivariant Polynomials

\(P(D(g) x) = D'(g) P(x)\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + 2(y-z) \\ z^4 + 100 x y z \\ z\end{bmatrix}\)

Not equivariant

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + y^2 + z^2 \end{bmatrix}\)

Equivariant

\(\begin{bmatrix} y\\ -x\\ z \end{bmatrix} \mapsto \begin{bmatrix} y^2 + 2(-x-z) \\ z^4 - 100 x y z\\ z\end{bmatrix}\)

\(R\)

\(?\)

(This one is actually invariant)

5 Tools to Build Equivariant Polynomials

  • 🔨 Composition \(\circ\)
  • 🔧 Addition \(+\)
  • 🔩 Multiplication \(\otimes\)
  • 💡 Linear Mixing

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + y^2 + z^2 \end{bmatrix}\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x\\y\\z \end{bmatrix}\)

5 Tools to Build Equivariant Polynomials

  • 🔨 Composition \(\circ\)
  • 🔧 Addition \(+\)
  • 🔩 Multiplication \(\otimes\)
  • 💡 Linear Mixing

Bottom-Up approach !!  Combine simple polynomials into more complex ones

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x^2 + y^2 + z^2 \end{bmatrix}\)

\(\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto \begin{bmatrix} x\\y\\z \end{bmatrix}\)

IrrepsArray
import e3nn_jax as e3nn
import e3nn_jax as e3nn


irreps = e3nn.Irreps("3x0e + 1o")
IrrepsArray
import e3nn_jax as e3nn


irreps = e3nn.Irreps("3x0e + 1o")

3 scalars

1 vector

IrrepsArray
import e3nn_jax as e3nn


irreps = e3nn.Irreps("3x0e + 1o")
array = jnp.array([0.0, 0.5, 0.5, 1.0, 2.0, 3.0])
IrrepsArray

3 scalars

1 vector

import e3nn_jax as e3nn


irreps = e3nn.Irreps("3x0e + 1o")
array = jnp.array([0.0, 0.5, 0.5, 1.0, 2.0, 3.0])


x = e3nn.IrrepsArray(irreps, array)
IrrepsArray

3 scalars

1 vector

🔨 Composition

two equivariant functions

\(f: V_1 \rightarrow V_2\)

\(h: V_2 \rightarrow V_3\)

\(h\circ f\) is equivariant!! 😊

\( h(f(D_1(g) x)) = h(D_2(g) f(x)) = D_3(g) h(f(x)) \)

🔨 Composition

two equivariant functions

\(f: V_1 \rightarrow V_2\)

\(h: V_2 \rightarrow V_3\)

\(h\circ f\) is equivariant!! 😊

def f(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

def h(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

# This composition is equivariant or the library raises an error!
h(f(x))

🔧 Addition

two equivariant functions

\(f: V_1 \rightarrow V_3\)

\(h: V_2 \rightarrow V_3\)

\(h + f\) is equivariant!! 😊

\( f(D_1(g) x) + h(D_2(g)x) = D_3(g) (f(x) + h(x)) \)

🔧 Addition

two equivariant functions

\(f: V_1 \rightarrow V_3\)

\(h: V_2 \rightarrow V_3\)

\(h + f\) is equivariant!! 😊

def f(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

def h(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

# This summation is equivariant or the library raises an error!
f(x) + h(x)

🔩 Multiplication

\(\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}\)

\(\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}\)

transforming with \(D(g)\)

transforming with \(D'(g)\)

\(= \begin{bmatrix} x_1y_1 & x_1y_2 & x_1y_3 & x_1y_4 & x_1y_5 \\ x_2y_1 & x_2y_2 & x_2y_3 & x_2y_4 & x_2y_5 \\ x_3y_1 & x_3y_2 & x_3y_3 & x_3y_4 & x_3y_5 \end{bmatrix}\)

🔩 Multiplication

\(\otimes\)

\(= \begin{bmatrix}{\color{red} x_1} {\color{blue} y_1}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} y_3}&{\color{red} x_1} {\color{blue} y_4}&{\color{red} x_1} {\color{blue} y_5}\\{\color{red} x_2} {\color{blue} y_1}&{\color{red} x_2} {\color{blue} y_2}&{\color{red} x_2} {\color{blue} y_3}&{\color{red} x_2} {\color{blue} y_4}&{\color{red} x_2} {\color{blue} y_5}\\{\color{red} x_3} {\color{blue} y_1}&{\color{red} x_3} {\color{blue} y_2}&{\color{red} x_3} {\color{blue} y_3}&{\color{red} x_3} {\color{blue} y_4}&{\color{red} x_3} {\color{blue} y_5}\end{bmatrix}\)

Tensor Product

transforms with \(D(g) \otimes D'(g)\) 😊

\(\dim( D \otimes D' ) = \dim( D ) \dim( D' )\)

\(\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}\)

\(\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}\)

Reducible representations

\(D\) defined on \(V\)

is reducible if

\(\exists W \subset V\)     \(W\neq0, V\)

such that

\(D|_W\) is a representation

Reducible representations

Famous Example

\(\begin{bmatrix}{\color{red} x_1} {\color{blue} x_2}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} z_2}\\{\color{red} y_1} {\color{blue} x_2}&{\color{red} y_1} {\color{blue} y_2}&{\color{red} y_1} {\color{blue} z_2}\\{\color{red} z_1} {\color{blue} x_2}&{\color{red} z_1} {\color{blue} y_2}&{\color{red} z_1} {\color{blue} z_2}\end{bmatrix}\)

\(\begin{bmatrix} {\color{red} x_1}\\{\color{red} y_1}\\{\color{red} z_1}\end{bmatrix}\otimes\begin{bmatrix} {\color{blue} x_2}\\{\color{blue} y_2}\\{\color{blue} z_2}\end{bmatrix} = \)

\(D\) defined on \(V\)

is reducible if

\(\exists W \subset V\)     \(W\neq0, V\)

such that

\(D|_W\) is a representation

Reducible representations

\({\color{red}x_1}{\color{blue}x_2} + {\color{red}y_1}{\color{blue}y_2} + {\color{red}z_1} {\color{blue}z_2}\)

\(\begin{bmatrix}c ( {\color{red}x_1} {\color{blue}z_2} + {\color{red}z_1} {\color{blue}x_2} ) \\ c ( {\color{red}x_1} {\color{blue}y_2} + {\color{red}y_1} {\color{blue}x_2} ) \\ 2 {\color{red}y_1} {\color{blue}y_2} - {\color{red}x_1} {\color{blue}x_2} - {\color{red}z_1} {\color{blue}z_2} \\ c ( {\color{red}y_1} {\color{blue}z_2} + {\color{red}z_1} {\color{blue}y_2} ) \\ c ( {\color{red}z_1} {\color{blue}z_2} - {\color{red}x_1} {\color{blue}x_2} ) \\\end{bmatrix}\)

\(\begin{bmatrix}{\color{red}y_1}{\color{blue}z_2}-{\color{red}z_1} {\color{blue}y_2}\\ {\color{red}z_1}{\color{blue}x_2}-{\color{red}x_1}{\color{blue}z_2}\\ {\color{red}x_1}{\color{blue}y_2}-{\color{red}y_1}{\color{blue}x_2}\end{bmatrix}\)

\(\begin{bmatrix}{\color{red} x_1} {\color{blue} x_2}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} z_2}\\{\color{red} y_1} {\color{blue} x_2}&{\color{red} y_1} {\color{blue} y_2}&{\color{red} y_1} {\color{blue} z_2}\\{\color{red} z_1} {\color{blue} x_2}&{\color{red} z_1} {\color{blue} y_2}&{\color{red} z_1} {\color{blue} z_2}\end{bmatrix}\)

\(3\times3=1+3+5\)

These can be seen as polynomials!

\(D\) is reducible if

\(\exists W \subset V\)

such that

\(D|_W\) is a representation

Irreducible representations

For the group of rotations (\(SO(3)\))

They are index by \(L=0, 1, 2, \dots\)

Of dimension \(2L+1\)

L=0 d=1 scalar, s orbital
L=1 d=3 vector, p orbital
L=2 d=5 d orbital
...

Irreducible representations

For the group of rotations (\(SO(3)\))

They are index by \(L=0, 1, 2, \dots\)

Of dimension \(2L+1\)

L=0 d=1 scalar, s orbital
L=1 d=3 vector, p orbital
L=2 d=5 d orbital
...

to use to achieve better data efficiency

Irreducible representations

For the group of rotations + parity (\(O(3)\))

They are index by \(L=0, 1, 2, \dots\)

and \(p=\pm 1\)

Of dimension \(2L+1\)

Even

Odd

L=0 d=1 scalar 0e
L=1 d=3 pseudo vector 1e
L=2 d=5 2e
...
L=0 d=1 pseudo scalar 0o
L=1 d=3 vector 1o
L=2 d=5 2o
...

Irreducible representations

For the group of rotations + parity (\(O(3)\))

They are index by \(L=0, 1, 2, \dots\)

and \(p=\pm 1\)

Of dimension \(2L+1\)

Even

Odd

L=0 d=1 scalar 0e
L=1 d=3 pseudo vector 1e
L=2 d=5 2e
...
L=0 d=1 pseudo scalar 0o
L=1 d=3 vector 1o
L=2 d=5 2o
...
e3nn.Irreps("0e")
e3nn.Irreps("1e")
e3nn.Irreps("2e")
e3nn.Irreps("3e")
# ...
e3nn.Irreps("0o")
e3nn.Irreps("1o")
e3nn.Irreps("2o")
e3nn.Irreps("3o")
# ...

🔩 Multiplication

\(L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)\)

🔩 Multiplication

\(L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)\)

generalization of \(3\times3=1+3+5\)

  • \( 2 \otimes 1 = 1 \oplus 2 \oplus 3 \)
  • \( 2 \otimes 2 = 0\oplus 1 \oplus 2 \oplus 3 \oplus 4 \)

🔩 Multiplication

\(L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)\)

  • \( 2 \otimes 1 = 1 \oplus 2 \oplus 3 \)
     
  • \( 2 \otimes 2 = 0\oplus 1 \oplus 2 \oplus 3 \oplus 4 \)
e3nn.Irrep("2e") * e3nn.Irrep("1o")
# [1o, 2o, 3o]

e3nn.Irrep("2e") * e3nn.Irrep("2o")
# [0o, 1o, 2o, 3o, 4o]

🔩 Multiplication

import e3nn_jax as e3nn


def f(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

def g(y: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial


def h(x: e3nn.IrrepsArray, y: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  return e3nn.tensor_product(f(x), g(y))

🔩 Multiplication

import e3nn_jax as e3nn


def f(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

def g(y: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial


def h(x: e3nn.IrrepsArray, y: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  return e3nn.tensor_product(f(x), g(y))

# symmetric
def f2(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  return e3nn.tensor_square(f(x))

🔩 Multiplication

import e3nn_jax as e3nn


def f(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial

def g(y: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  # Equivariant Polynomial


def h(x: e3nn.IrrepsArray, y: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  return e3nn.tensor_product(f(x), g(y))

# symmetric degree 2
def f2(x: e3nn.IrrepsArray) -> e3nn.IrrepsArray:
  return e3nn.tensor_square(f(x))

# symmetric degree n
cgs = reduced_symmetric_tensor_product_basis(irreps, n)

💡 Linear Mixing

3 scalars

a vector

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

a vector

3 scalars

a vector

\(\begin{bmatrix} b^1\\b^2\\b^3\\b^4\\b^5\\b^6\\b^7\\b^8\\b^9\end{bmatrix}\)

a vector

Linear map

💡 Linear Mixing

3 scalars

a vector

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

a vector

3 scalars

a vector

\(\begin{bmatrix} b^1\\b^2\\b^3\\b^4\\b^5\\b^6\\b^7\\b^8\\b^9\end{bmatrix}\)

a vector

\(w_1\)

\(w_2\)

\(w_3\)

by Schur's lemma

💡 Linear Mixing

3 scalars

a vector

\(\begin{bmatrix} a^1\\a^2\\a^3\\a^4\\a^5\\a^6\\a^7\\a^8\\a^9\end{bmatrix}\)

a vector

3 scalars

a vector

\(\begin{bmatrix} b^1\\b^2\\b^3\\b^4\\b^5\\b^6\\b^7\\b^8\\b^9\end{bmatrix}\)

a vector

\(w_1\)

\(w_2\)

\(w_3\)

import e3nn_jax as e3nn

a = e3nn.IrrepsArray("3x0e + 2x1o", jnp.array([a1, a2, a3, a4, a5, a6, a7, a8, a9]))

lin = e3nn.flax.Linear("1o + 3x0e + 1o")
w = lin.init(seed, a)

b = lin.apply(w, a)

🔨 Composition

🔧 Addition

🔩 Multiplication

💡 Linear Mixing

Spherical Harmonics

def spherical_harmonics(l: int, x):
    # Check that x is a vector
    assert x.irreps == "1o"

    # Output representation
    irrep_out = e3nn.Irrep(l, (-1) ** l)

    if l == 0:
        return e3nn.IrrepsArray(irrep_out, jnp.array([1.0]))

    y = spherical_harmonics(l - 1, x)
    return e3nn.tensor_product(y, x).filter(keep=irrep_out)

# Test
x = e3nn.IrrepsArray("1o", jnp.array([0.508, 0.816, -0.408]))
spherical_harmonics(5, x)

# 1x5o
# [-0.02850328  0.04899025  0.11198934 -0.28012297  0.02936267 -0.18379876
#  -0.0235826  -0.06189997  0.22650346 -0.10543777  0.0070266 ]

Conclusion

  • Group and representations
  • Equivariance \(f({\color{purple}D(g)} x) = {\color{darkgreen}D'(g)} f(x)\)
    • 🔨 Composition
    • 🔧 Addition
  • Polynomials (Bottom-Up)
    •  🔩 Tensor Product  \(L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)\)
    • 💡 Linear Mixing
  • ​Example
    • ​Spherical Harmonics

Thank you for listening!

Boston Symmetry Day

By Mario Geiger

Boston Symmetry Day

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