Learning Geometry
&
3D Symmetries
Mario Geiger
This morning Emmanuel Noutahi presented us the different molecular representations
This morning Emmanuel Noutahi presented us the different molecular representations
Equivariant Neural Networks
input
output
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],
[-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]])Why we want Equivariance?
array([[ 0.28261471, 0.56535286, 1.38205716],
[-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
Fluid dynamics
Mechanics
Electrodynamics
Standard Model
Rotation
Translation
Boosts
(Galilean or Lorentz)
Time
translation
Example of Equivariance
\(f:\) positions \(\to\) forces
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
Cosmology, Medical Images and others
44M atoms while taking advantage of up to 5120 GPUs
Albert Musaelian
We observe: Equivariance \(\Rightarrow\) Data Efficient!
(Nequip: Simon Batzner et al. 2021)
max L of the messages
Error
Trainset size
Invariant features
(Nequip: Simon Batzner et al. 2021)
Equivariant features
Error
Trainset size
Invariant features
We observe: Equivariance \(\Rightarrow\) Data Efficient!
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
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
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 independently
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
\(V\)
\(V'\)
\(D(g)\)
\(D'(g)\)
\(V\)
\(V'\)
\(f\)
\(f\)
\(f(D(g) x)\)
\(D'(g) f(x)\)
\(=\)
Fluid dynamics
Mechanics
Electrodynamics
Standard Model
Rotation
Translation
Boosts
(Galilean or Lorentz)
These models are defined as invariant lagrangians \(\mathcal L(\text{state})\)
\(\mathcal L(D(g, \text{state})) = \mathcal L(\text{state})\)
Time
translation
🔨 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!
\( f(D_1(g) x) + h(D_2(g)x) = D_3(g) (f(x) + h(x)) \)
equivariant function: \(f: V_1 \rightarrow V_2\)
a scalar: \(\alpha \in \mathbb{R}\)
\(\alpha f\) is equivariant!
\( \alpha f(D_1(g) x) = D_2(g) \alpha f(x) \)
🔩 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 \(D(g)\)
transforming with \(D'(g)\)
\(= \begin{bmatrix} x_1y_1 & x_1y_2 & x_1y_3 & x_1y_4 & x_1y_5 \\ x_2y_1 & x_2y_2 & x_2y_3 & x_2y_4 & x_2y_5 \\ x_3y_1 & x_3y_2 & x_3y_3 & x_3y_4 & x_3y_5 \end{bmatrix}\)
🔩 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 \(D(g) \otimes D'(g)\) 👍
\(\dim( D \otimes D' ) = \dim( D ) \dim( D' )\) 👎
\(\begin{bmatrix} {\color{red} x_1}\\{\color{red} x_2}\\{\color{red} x_3}\end{bmatrix}\)
\(\begin{bmatrix} {\color{blue} y_1}\\{\color{blue} y_2}\\{\color{blue} y_3}\\{\color{blue} y_4}\\{\color{blue} y_5} \end{bmatrix}\)
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
\({\color{red}x_1}{\color{blue}x_2} + {\color{red}y_1}{\color{blue}y_2} + {\color{red}z_1} {\color{blue}z_2}\)
\(\begin{bmatrix}c ( {\color{red}x_1} {\color{blue}z_2} + {\color{red}z_1} {\color{blue}x_2} ) \\ c ( {\color{red}x_1} {\color{blue}y_2} + {\color{red}y_1} {\color{blue}x_2} ) \\ 2 {\color{red}y_1} {\color{blue}y_2} - {\color{red}x_1} {\color{blue}x_2} - {\color{red}z_1} {\color{blue}z_2} \\ c ( {\color{red}y_1} {\color{blue}z_2} + {\color{red}z_1} {\color{blue}y_2} ) \\ c ( {\color{red}z_1} {\color{blue}z_2} - {\color{red}x_1} {\color{blue}x_2} ) \\\end{bmatrix}\)
\(\begin{bmatrix}{\color{red}y_1}{\color{blue}z_2}-{\color{red}z_1} {\color{blue}y_2}\\ {\color{red}z_1}{\color{blue}x_2}-{\color{red}x_1}{\color{blue}z_2}\\ {\color{red}x_1}{\color{blue}y_2}-{\color{red}y_1}{\color{blue}x_2}\end{bmatrix}\)
\(\begin{bmatrix}{\color{red} x_1} {\color{blue} x_2}&{\color{red} x_1} {\color{blue} y_2}&{\color{red} x_1} {\color{blue} z_2}\\{\color{red} y_1} {\color{blue} x_2}&{\color{red} y_1} {\color{blue} y_2}&{\color{red} y_1} {\color{blue} z_2}\\{\color{red} z_1} {\color{blue} x_2}&{\color{red} z_1} {\color{blue} y_2}&{\color{red} z_1} {\color{blue} z_2}\end{bmatrix}\)
\(3\times3=1+3+5\)
\(D\) defined on \(V\)
is reducible if
\(\exists W \subset V\) \(W\neq0, V\)
such that
\(D|_W\) is a representation
Reducible representations
\(D\) defined on \(V\)
is irreducible if
only for \(W = 0\) or \(W=V\)
\(D|_W\) is a representation
Irreducible
Irreducible representations
For the group of rotations (\(SO(3)\))
They are index by \(L=0, 1, 2, \dots\)
Of dimension \(d=2L+1\)
| L=0 | d=1 | scalar |
| L=1 | d=3 | vector |
| L=2 | d=5 | |
| ... |
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 \(d=2L+1\)
Even: \(p=1\)
Odd: \(p=-1\)
| L=0 | d=1 | scalar |
| L=1 | d=3 | pseudo vector |
| L=2 | d=5 | |
| ... |
| L=0 | d=1 | pseudo scalar |
| L=1 | d=3 | vector |
| L=2 | d=5 | |
| ... |
🔩 Tensor Product
\(L_1 \otimes L_2 = |L_1-L_2| \oplus \dots \oplus (L_1+L_2)\)
Clebsch-Gordan Theorem
Tells you how to decompose the tensor product of two irreps into irreps
💡 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
🔨 Composition
🔧 Basic Arithmetic
🔩 Tensor Product
💡 Linear Mixing
🔨 Composition
🔧 Basic Arithmetic
🔩 Tensor Product
💡 Linear Mixing
-
Graph Message Passing
-
Transformer
-
3D Convolution
-
...
Thanks for your Attention!
Data-Efficiency
Protein Folding
Molecular Dynamics
Phonons
Quantum Physics
Docking
ML for Drug Discovery Summer School
By Mario Geiger
