Equivariant
Neural Networks
Mario Geiger
Equivariant Neural Networks
input
geometric object
output
geometric properties
Illustration of a neural network equivariant to rotations in 3D
Why we want Equivariance?
Why we want Equivariance?
array([[ 0.28261471, 0.56535286, 1.38205716],
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[-0.96598984, 0.06802525, 0.77853411],
[ 1.09923518, 1.20586634, 0.92461881],
[-0.35728519, 0.55409651, 0.3251024 ],
[-0.03344675, -0.48225385, -0.294099 ],
[-1.79362192, 1.26634314, -0.27039329]])array([[ 0.55856046, 1.19226885, 0.60782165],
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[ 0.27944288, -0.59813412, -0.24272213],
[ 2.40641258, 0.25619413, -1.19271872]])Why we want Equivariance?
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[-0.59397486, 0.04869514, 0.76054154],
[-0.96598984, 0.06802525, 0.77853411],
[ 1.09923518, 1.20586634, 0.92461881],
[-0.35728519, 0.55409651, 0.3251024 ],
[-0.03344675, -0.48225385, -0.294099 ],
[-1.79362192, 1.26634314, -0.27039329]])array([[ 0.55856046, 1.19226885, 0.60782165],
[ 0.59452502, 0.65252289, -0.4547258 ],
[ 0.72974089, 0.75212044, -0.7877266 ],
[ 1.00189192, 0.76199152, 1.55897624],
[ 1.10428859, 0.32784872, -0.08623213],
[ 0.27944288, -0.59813412, -0.24272213],
[ 2.40641258, 0.25619413, -1.19271872]])Data-Augmentation
- Inexact
- Expensive
Equivariance
- Exact
- Data-efficient
Example of Equivariance
\(f:\) positions \(\to\) forces
MLIP
We observe: Equivariance \(\Rightarrow\) Data Efficient!
Fluid dynamics
Mechanics
Electrodynamics
Standard Model
Rotation
Translation
Boosts
(Galilean or Lorentz)
Time
translation
Example of Equivariance
\(f:\) positions \(\to\) Hamiltonian
Where is equivariance used in AI?
Protein Folding
EquiFold Jae Hyeon Lee et al.
Protein Docking
DIFFDOCK Gabriele Corso et al.
Molecular Dynamics
Allegro A. Musaelian et al.
Open Catalyst Project.
Solid State Physics
Prediction of Phonon Density Z. Chen et al.
Molecular Electron Densities
Cracking the Quantum Scaling Limit with Machine Learned Electron Densities J. Rackers
Medical Images, Robotics, ...
44M atoms while taking advantage of up to 5120 GPUs
Albert Musaelian
Equivariant Tasks
Tasks
learn 3d object
from
2d images
Non-Equivariant
Expressivity
Equivariant
Model
Equivariant Tasks
Tasks
Non-Equivariant
Expressivity
Equivariant
Model
Math Slides
Equivariance
Group
Representation
Tools
- Composition
- Addition
- Amplification
- Tensor Product
- Linear mixing
\(\rightarrow\) is my task equivariant?
\(\rightarrow\) to make my model equivariant
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 group acts linearly"
Group and Representations
"what are the operations"
"how they compose"
examples:
rotations
lorentz group
examples:
scalars, vectors, pseudovectors, ...
scalars, 4-vector
"vector spaces on which the group acts linearly"
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Representation \((V, D)\)
- \(g\in G\), \(x \in V\) \(x \mapsto D(g,x)\)
- Linearity \(D(g, x+y) = D(g,x) + D(g,y) \)
- Follow the structure of the group
\(D(g\cdot h,x) = D(g, D(h,x))\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Representation \((V, D)\)
- \(g\in G\), \(x \in V\) \(x \mapsto D(g,x)\)
- Linearity \(D(g, x+y) = D(g,x) + D(g,y) \)
- Follow the structure of the group
\(D(g\cdot h,x) = D(g, D(h,x))\)
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Representation \((V, D)\)
- \(g\in G\), \(x \in V\) \(x \mapsto D(g,x)\)
- Linearity \(D(g, x+y) = D(g,x) + D(g,y) \)
- Follow the structure of the group
\(D(g\cdot h,x) = D(g, D(h,x))\)
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Representation \((V, D)\)
- \(g\in G\), \(x \in V\) \(x \mapsto D(g,x)\)
- Linearity \(D(g, x+y) = D(g,x) + D(g,y) \)
- Follow the structure of the group
\(D(g\cdot h,x) = D(g, D(h,x))\)
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
Group and Representations
"what are the operations" "how they compose"
"vector spaces on which the group acts linearly"
Representation \((V, D)\)
- \(g\in G\), \(x \in V\) \(x \mapsto D(g,x)\)
- Linearity \(D(g, x+y) = D(g,x) + D(g,y) \)
- Follow the structure of the group
\(D(g\cdot h,x) = D(g, D(h,x))\)
Group \(G\)
- \(\text{identity} \in G\)
- associativity \(g \cdot (h \cdot k) = (g \cdot h) \cdot k\)
- inverse \(g^{-1} \in G\)
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
\(V\)
\(V'\)
\(D(g)\)
\(D'(g)\)
\(V\)
\(V'\)
\(f\)
\(f\)
\(f(D(g, x))\)
\(D'(g, f(x))\)
\(=\)
Equivariant Tasks
Tasks
\(f(D(g, x))\)
\(D'(g, f(x))\)
\(=\)
\(x\)
\(f(x)\)
🔨 Composition
🔧 Basic Arithmetic
🔩 Tensor Product
💡 Linear Mixing
🔨 Composition
two equivariant functions
\(f: V_1 \rightarrow V_2\)
\(h: V_2 \rightarrow V_3\)
\(h\circ f\) is equivariant!
\( h(f(\ D_1(g, x)\ )) = h(\ D_2(g, f(x))\ ) = D_3(g, h(f(x))) \)
🔧 Basic Arithmetic \(+-*/\)
two equivariant functions
\(f: V_1 \rightarrow V_3\)
\(h: V_2 \rightarrow V_3\)
\(h + f\) is equivariant!
\( f(D_1(g,x)) + h(D_2(g,x)) = D_3(g, f(x) + h(x)) \)
🔧 Basic Arithmetic \(+-*/\)
two equivariant functions
\(f: V_1 \rightarrow V_3\)
\(h: V_2 \rightarrow V_3\)
\(h + f\) is equivariant!
equivariant function: \(f: V_1 \rightarrow V_2\)
a scalar: \(\alpha \in \mathbb{R}\)
\(\alpha f\) is equivariant!
\( \alpha f(D_1(g,x)) = D_2(g, \alpha f(x)) \)
\( f(D_1(g,x)) + h(D_2(g,x)) = D_3(g, f(x) + h(x)) \)
🔩 Tensor Product
\(\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}\)
\(\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}\)
transforming with \(D(g, \cdot)\)
transforming with \(D'(g, \cdot)\)
\(= \begin{bmatrix} x_1y_1 & x_1y_2 & x_1y_3 & x_1y_4 & x_1y_5 \\ x_2y_1 & x_2y_2 & x_2y_3 & x_2y_4 & x_2y_5 \\ x_3y_1 & x_3y_2 & x_3y_3 & x_3y_4 & x_3y_5 \end{bmatrix}\)
🔩 Tensor Product
\(\otimes\)
\(= \begin{bmatrix}{\color{red} x_1} {\color{blue} y_1}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} y_3}&{\color{red} x_1} {\color{blue} y_4}&{\color{red} x_1} {\color{blue} y_5}\\{\color{red} x_2} {\color{blue} y_1}&{\color{red} x_2} {\color{blue} y_2}&{\color{red} x_2} {\color{blue} y_3}&{\color{red} x_2} {\color{blue} y_4}&{\color{red} x_2} {\color{blue} y_5}\\{\color{red} x_3} {\color{blue} y_1}&{\color{red} x_3} {\color{blue} y_2}&{\color{red} x_3} {\color{blue} y_3}&{\color{red} x_3} {\color{blue} y_4}&{\color{red} x_3} {\color{blue} y_5}\end{bmatrix}\)
transforms with a representation called \(D\otimes D'\)
\(\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}\)
\(\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}\)
💡 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
\(w_1\)
\(w_2\)
\(w_3\)
The Schur's lemma gives you the linear maps between two representations
🔨 Composition
🔧 Basic Arithmetic
🔩 Tensor Product
💡 Linear Mixing
Graph Message Passing
Transformer
3D Convolution
etc
🔨 Composition
🔧 Basic Arithmetic
🔩 Tensor Product
💡 Linear Mixing
Graph Message Passing
Transformer
3D Convolution
etc
🔨 Composition
🔧 Basic Arithmetic
🔩 Tensor Product
💡 Linear Mixing
Graph Message Passing
Transformer
3D Convolution
etc
Protein Folding
Molecular Dynamics
Phonons
Docking
Medical Imagery
Data-Augmentation
- Inexact
- Expensive
Equivariance
- Exact
- Data-efficient
task
model
equiv.
equiv.