Positional Encoding
Positional Encoding Methods
GeometricFlux.AbstractPositionalEncoding — TypeAbstractPositionalEncodingAbstract type of positional encoding for GNN.
GeometricFlux.RandomWalkPE — TypeRandomWalkPE{K}Concrete type of positional encoding from random walk method.
See also positional_encode for generating positional encoding.
GeometricFlux.LaplacianPE — TypeLaplacianPE{K}Concrete type of positional encoding from graph Laplacian method.
See also positional_encode for generating positional encoding.
GeometricFlux.positional_encode — Functionpositional_encode(RandomWalkPE{K}, A)Returns positional encoding (PE) of size (K, N) where N is node number. PE is generated by K-step random walk over given graph.
Arguments
K::Int: First dimension of PE.A: Adjacency matrix of a graph.
See also RandomWalkPE for random walk method.
positional_encode(LaplacianPE{K}, A)Returns positional encoding (PE) of size (K, N) where N is node number. PE is generated from eigenvectors of a graph Laplacian truncated by K.
Arguments
K::Int: First dimension of PE.A: Adjacency matrix of a graph.
See also LaplacianPE for graph Laplacian method.
Positional Encoding Layers
$E(n)$-equivariant Positional Encoding Layer
It employs message-passing scheme and can be defined by following functions:
- message function: $y_{ij}^l = (x_i^l - x_j^l)\phi_x(m_{ij})$
- aggregate: $y_i^l = \frac{1}{M} \sum_{j \in \mathcal{N}(i)} y_{ij}^l$
- update function: $x_i^{l+1} = x_i^l + y_i^l$
where $x_i^l$ and $x_j^l$ denote the positional feature for node $i$ and $j$, respectively, in $l$-th layer, $\phi_x$ is the neural network for positional encoding and $m_{ij}$ is the edge feature for edge $(i,j)$. $y_{ij}^l$ and $y_i^l$ represent the encoded positional feature and aggregated positional feature, respectively, and $M$ denotes number of neighbors of node $i$.
GeometricFlux.EEquivGraphPE — TypeEEquivGraphPE(in_dim=>out_dim; init=glorot_uniform, bias=true)E(n)-equivariant positional encoding layer.
Arguments
in_dim::Int: dimension of input positional feature.out_dim::Int: dimension of output positional feature.init: neural network initialization function.bias::Bool: dimension of edge feature.
Examples
julia> in_dim_edge, out_dim = 2, 5
(2, 5)
julia> l = EEquivGraphPE(in_dim_edge=>out_dim)
EEquivGraphPE(2 => 5)See also EEquivGraphConv.
Reference: Victor Garcia Satorras, Emiel Hoogeboom, Max Welling (2021)
Learnable Structural Positional Encoding layer
(WIP)
GeometricFlux.LSPE — TypeLSPE(fg, f_h, f_e, f_p, k; pe_method=RandomWalkPE)Learnable structural positional encoding layer. LSPE layer can be seen as a GNN layer warpped in WithGraph.
Arguments
fg::FeaturedGraph: A given graoh for positional encoding.f_h::MessagePassing: Neural network layer for node update.f_e: Neural network layer for edge update.f_p: Neural network layer for positional encoding.k::Int: Dimension of positional encoding.pe_method: Initializer for positional encoding.
Reference: Vijay Prakash Dwivedi, Anh Tuan Luu, Thomas Laurent, Yoshua Bengio, Xavier Bresson (2021)