add LSPE

docs/bibliography.bib

@@ -197,3 +197,12 @@ @inproceedings{Satorras2021
    url = {http://arxiv.org/abs/2102.09844},
    year = {2021},
 }
+
+@article{Dwivedi2021,
+   abstract = {Graph neural networks (GNNs) have become the standard learning architectures for graphs. GNNs have been applied to numerous domains ranging from quantum chemistry, recommender systems to knowledge graphs and natural language processing. A major issue with arbitrary graphs is the absence of canonical positional information of nodes, which decreases the representation power of GNNs to distinguish e.g. isomorphic nodes and other graph symmetries. An approach to tackle this issue is to introduce Positional Encoding (PE) of nodes, and inject it into the input layer, like in Transformers. Possible graph PE are Laplacian eigenvectors. In this work, we propose to decouple structural and positional representations to make easy for the network to learn these two essential properties. We introduce a novel generic architecture which we call LSPE (Learnable Structural and Positional Encodings). We investigate several sparse and fully-connected (Transformer-like) GNNs, and observe a performance increase for molecular datasets, from 2.87% up to 64.14% when considering learnable PE for both GNN classes.},
+   author = {Vijay Prakash Dwivedi and Anh Tuan Luu and Thomas Laurent and Yoshua Bengio and Xavier Bresson},
+   month = {10},
+   title = {Graph Neural Networks with Learnable Structural and Positional Representations},
+   url = {http://arxiv.org/abs/2110.07875},
+   year = {2021},
+}

docs/src/manual/positional.md

@@ -1,6 +1,17 @@
-# Positional Encoding Layers
+# Positional Encoding
 
-## ``E(n)``-equivariant Positional Encoding Layer
+## Positional Encoding Methods
+
+```@docs
+AbstractPositionalEncoding
+RandomWalkPE
+LaplacianPE
+positional_encode
+```
+
+## Positional Encoding Layers
+
+### ``E(n)``-equivariant Positional Encoding Layer
 
 It employs message-passing scheme and can be defined by following functions:
 
@@ -17,3 +28,16 @@ EEquivGraphPE
 Reference: [Satorras2021](@cite)
 
 ---
+
+### Learnable Structural Positional Encoding layer
+
+(WIP)
+
+```@docs
+LSPE
+```
+
+Reference: [Dwivedi2021](@cite)
+
+---
+

src/GeometricFlux.jl

@@ -32,9 +32,12 @@ export
     MessagePassing,
 
     # layers/positional
-    AbstractPE,
+    AbstractPositionalEncoding,
+    RandomWalkPE,
+    LaplacianPE,
     positional_encode,
     EEquivGraphPE,
+    LSPE,
 
     # layers/graph_conv
     GCNConv,

src/layers/positional.jl

@@ -1,11 +1,81 @@
 """
-    AbstractPE
+    AbstractPositionalEncoding
 
 Abstract type of positional encoding for GNN.
 """
-abstract type AbstractPE end
+abstract type AbstractPositionalEncoding end
+
+"""
+    RandomWalkPE{K}
+
+Concrete type of positional encoding from random walk method.
+
+See also [`positional_encode`](@ref) for generating positional encoding.
+"""
+struct RandomWalkPE{K} <: AbstractPositionalEncoding end
+
+"""
+    LaplacianPE{K}
+
+Concrete type of positional encoding from graph Laplacian method.
+
+See also [`positional_encode`](@ref) for generating positional encoding.
+"""
+struct LaplacianPE{K} <: AbstractPositionalEncoding end
+
+"""
+    positional_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`](@ref) for random walk method.
+"""
+function positional_encode(::Type{RandomWalkPE{K}}, A::AbstractMatrix) where {K}
+    N = size(A, 1)
+    @assert K ≤ N "K=$K must less or equal to number of nodes ($N)"
+    inv_D = GraphSignals.degree_matrix(A, Float32, inverse=true)
+
+    RW = similar(A, size(A)..., K)
+    RW[:, :, 1] .= A * inv_D
+    for i in 2:K
+        RW[:, :, i] .= RW[:, :, i-1] * RW[:, :, 1]
+    end
+
+    pe = similar(RW, K, N)
+    for i in 1:N
+        pe[:, i] .= RW[i, i, :]
+    end
+
+    return pe
+end
+
+"""
+    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`](@ref) for graph Laplacian method.
+"""
+function positional_encode(::Type{LaplacianPE{K}}, A::AbstractMatrix) where {K}
+    N = size(A, 1)
+    @assert K ≤ N "K=$K must less or equal to number of nodes ($N)"
+    L = GraphSignals.normalized_laplacian(A)
+    U = eigvecs(L)
+    return U[1:K, :]
+end
 
-positional_encode(l::AbstractPE, args...) = throw(ErrorException("positional_encode function for $l is not implemented."))
 
 """
     EEquivGraphPE(in_dim=>out_dim; init=glorot_uniform, bias=true)
@@ -78,3 +148,77 @@ output_dim(l::EEquivGraphPE) = size(l.nn.weight, 1)
 
 positional_encode(wg::WithGraph{<:EEquivGraphPE}, args...) = wg(args...)
 positional_encode(l::EEquivGraphPE, args...) = l(args...)
+
+"""
+    LSPE(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.
+"""
+struct LSPE{H<:MessagePassing,E,F,P} <: AbstractPositionalEncoding
+    f_h::H
+    f_e::E
+    f_p::F
+    pe::P
+end
+
+function LSPE(fg::AbstractFeaturedGraph, f_h::MessagePassing, f_e, f_p, k::Int;
+              pe_method=RandomWalkPE)
+    A = GraphSignals.adjacency_matrix(fg)
+    return LSPE(f_h, f_e, f_p, positional_encode(pe_method{k}, A))
+end
+
+# For variable graph
+function (l::LSPE)(fg::AbstractFeaturedGraph)
+    X = node_feature(fg)
+    E = edge_feature(fg)
+    GraphSignals.check_num_nodes(fg, X)
+    GraphSignals.check_num_edges(fg, E)
+    E, V = propagate(l, graph(fg), E, X)
+    return ConcreteFeaturedGraph(fg, nf=V, ef=E)
+end
+
+# For static graph
+function (l::LSPE)(el::NamedTuple, X::AbstractArray, E::AbstractArray)
+    GraphSignals.check_num_nodes(el.N, X)
+    GraphSignals.check_num_edges(el.E, E)
+    E, V = propagate(l, graph(fg), E, X)
+    return V, E
+end
+
+update_vertex(l::LSPE, el::NamedTuple, X, E::AbstractArray) = l.f_h(el, X, E)
+update_vertex(l::LSPE, el::NamedTuple, X, E::Nothing) = l.f_h(el, X)
+
+update_edge(l::LSPE, h_i, h_j, e_ij) = l.f_e(e_ij)
+
+positional_encode(l::LSPE, p_i, p_j, e_ij) = l.f_p(p_i)
+
+propagate(l::LSPE, sg::SparseGraph, E, V) = propagate(l, to_namedtuple(sg), E, V)
+
+function propagate(l::LSPE, el::NamedTuple, E, V)
+    e_ij = _gather(E, el.es)
+    h_i = _gather(V, el.xs)
+    h_j = _gather(V, el.nbrs)
+    p_i = _gather(l.pe, el.xs)
+    p_j = _gather(l.pe, el.nbrs)
+
+    V = update_vertex(l, el, vcat(V, l.pe), E)
+    E = update_edge(l, h_i, h_j, e_ij)
+    l.pe = positional_encode(l, p_i, p_j, e_ij)
+    return E, V
+end
+
+function Base.show(io::IO, l::LSPE)
+    print(io, "LSPE(node_layer=", l.f_h)
+    print(io, ", edge_layer=", l.f_e)
+    print(io, ", positional_encode=", l.f_p, ")")
+end

test/layers/positional.jl

@@ -38,4 +38,17 @@
             @test length(g.grads) == 2
         end
     end
+
+    @testset "LSPE" begin
+        K = 3
+        f_h = GraphConv(in_channel=>out_channel)
+        f_e = Dense(in_channel, out_channel)
+        f_p = Dense(in_channel, out_channel)
+        l = LSPE(fg, f_h, f_e, f_p, K)
+
+        # nf = rand(T, out_channel, N)
+        # fg = FeaturedGraph(adj, nf=nf)
+        # fg_ = l(fg)
+        # @test size(node_feature(fg_)) == (out_channel, N)
+    end
 end