Add EEquivGraphPE layer and introduce nested EEquivGraphConv

Project.toml

@@ -32,7 +32,7 @@ DataStructures = "0.18"
 FillArrays = "0.13"
 Flux = "0.12 - 0.13"
 GraphMLDatasets = "0.1"
-GraphSignals = "0.4 - 0.5"
+GraphSignals = "0.6"
 Graphs = "1"
 NNlib = "0.8"
 NNlibCUDA = "0.2"

docs/bibliography.bib

@@ -183,3 +183,17 @@ @inproceedings{Hamilton2017
    title = {Inductive Representation Learning on Large Graphs},
    year = {2017},
 }
+
+@inproceedings{Satorras2021,
+   abstract = {This paper introduces a new model to learn graph neural networks equivariant to rotations, translations, reflections and permutations called E(n)-Equivariant Graph Neural Networks (EGNNs). In contrast with existing methods, our work does not require computationally expensive higher-order representations in intermediate layers while it still achieves competitive or better performance. In addition, whereas existing methods are limited to equivariance on 3 dimensional spaces, our model is easily scaled to higher-dimensional spaces. We demonstrate the effectiveness of our method on dynamical systems modelling, representation learning in graph autoencoders and predicting molecular properties.},
+   author = {Victor Garcia Satorras and Emiel Hoogeboom and Max Welling},
+   editor = {Marina Meila and Tong Zhang},
+   booktitle = {Proceedings of the 38th International Conference on Machine Learning},
+   month = {2},
+   pages = {9323-9332},
+   publisher = {PMLR},
+   title = {E(n) Equivariant Graph Neural Networks},
+   volume = {139},
+   url = {http://arxiv.org/abs/2102.09844},
+   year = {2021},
+}

docs/make.jl

@@ -42,8 +42,10 @@ makedocs(
              "Dynamic Graph Update" => "dynamicgraph.md",
              "Manual" => [
                "FeaturedGraph" => "manual/featuredgraph.md",
-               "Graph Convolutional Layers" => "manual/conv.md",
+               "Graph Convolutional Layers" => "manual/graph_conv.md",
                "Graph Pooling Layers" => "manual/pool.md",
+               "Group Convolutional Layers" => "manual/group_conv.md",
+               "Positional Encoding Layers" => "manual/positional.md",
                "Embeddings" => "manual/embedding.md",
                "Models" => "manual/models.md",
                "Linear Algebra" => "manual/linalg.md",

docs/src/manual/featuredgraph.md

@@ -9,6 +9,8 @@ GraphSignals.edge_feature
 GraphSignals.has_edge_feature
 GraphSignals.global_feature
 GraphSignals.has_global_feature
+GraphSignals.positional_feature
+GraphSignals.has_positional_feature
 GraphSignals.subgraph
 GraphSignals.ConcreteFeaturedGraph
 ```

docs/src/manual/group_conv.md

@@ -0,0 +1,19 @@
+# Group Convolutional Layers
+
+## ``E(n)``-equivariant Convolutional Layer
+
+It employs message-passing scheme and can be defined by following functions:
+
+- message function (Eq. 3 from paper): ``m_{ij} = \phi_e(h_i^l, h_j^l, ||x_i^l - x_j^l||^2, a_{ij})``
+- aggregate (Eq. 5 from paper): ``m_i = \sum_j m_{ij}``
+- update function (Eq. 6 from paper): ``h_i^{l+1} = \phi_h(h_i^l, m_i)``
+
+where ``h_i^l`` and ``h_j^l`` denotes the node feature for node ``i`` and ``j``, respectively, in ``l``-th layer, as well as ``x_i^l`` and ``x_j^l`` denote the positional feature for node ``i`` and ``j``, respectively, in ``l``-th layer. ``a_{ij}`` is the edge feature for edge ``(i,j)``. ``\phi_e`` and ``\phi_h`` are neural network for edges and nodes.
+
+```@docs
+EEquivGraphConv
+```
+
+Reference: [Satorras2021](@cite)
+
+---

docs/src/manual/positional.md

@@ -0,0 +1,19 @@
+# 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``.
+
+```@docs
+EEquivGraphPE
+```
+
+Reference: [Satorras2021](@cite)
+
+---

src/GeometricFlux.jl

@@ -31,6 +31,11 @@ export
     # layers/msgpass
     MessagePassing,
 
+    # layers/positional
+    AbstractPE,
+    positional_encode,
+    EEquivGraphPE,
+
     # layers/graph_conv
     GCNConv,
     ChebConv,
@@ -75,6 +80,7 @@ include("layers/graphlayers.jl")
 include("layers/gn.jl")
 include("layers/msgpass.jl")
 
+include("layers/positional.jl")
 include("layers/graph_conv.jl")
 include("layers/group_conv.jl")
 include("layers/pool.jl")

src/groups.jl

@@ -0,0 +1,28 @@
+import Base: *, inv
+
+struct R{N} end
+
+Base.ndims(::R{N}) where {N} = N
+
+Base.identity(::R{N}, ::Type{T}=Float32) where {N,T<:Number} = zeros(T, N)
+
+
+struct H{N} end
+
+Base.ndims(::H{N}) where {N} = N
+
+Base.identity(::H{N}, ::Type{T}=Float32) where {N,T<:Number} = ones(T, N)
+
+(*)(h1, h2) = h1 * h2
+
+inv(h) = 1. / h
+
+Base.log(h) = log.(h)
+
+Base.exp(c) = exp.(c)
+
+"""
+The logarithmic distance ||log(inv(h1).h2)||
+
+"""
+dist(h1, h2) = log(inv(h1) * h2)

src/layers/group_conv.jl

@@ -1,128 +1,110 @@
 """
-    EEquivGraphConv(in_dim, int_dim, out_dim; init=glorot_uniform)
-    EEquivGraphConv(in_dim, nn_edge, nn_x, nn_h)
+    EEquivGraphConv(in_dim=>out_dim, pos_dim, edge_dim; init=glorot_uniform)
 
-E(n)-equivariant graph neural network layer as defined in the paper "[E(n) Equivariant Neural Networks](https://arxiv.org/abs/2102.09844)" by Satorras, Hoogeboom, and Welling (2021).
+E(n)-equivariant graph neural network layer.
 
 # Arguments
 
-Either one of two sets of arguments:
-
-Set 1:
-
-- `in_dim`: node feature dimension. Data is assumed to be of the form [feature; coordinate], so `in_dim` must strictly be less than the dimension of the input vectors.
-- `int_dim`: intermediate dimension, can be arbitrary.
+- `in_dim::Int`: node feature dimension. Data is assumed to be of the form [feature; coordinate], so `in_dim` must strictly be less than the dimension of the input vectors.
 - `out_dim`: the output of the layer will have dimension `out_dim` + (dimension of input vector - `in_dim`).
-- `init`: neural network initialization function, should be compatible with `Flux.Dense`.
-
-Set 2:
+- `pos_dim::Int`: dimension of positional encoding.
+- `edge_dim::Int`: dimension of edge feature.
+- `init`: neural network initialization function.
 
-- `in_dim`: as in Set 1.
-- `nn_edge`: a differentiable function that must take vectors of dimension `in_dim * 2 + 2` (output designated `int_dim`)
-- `nn_x`: a differentiable function that must take vectors of dimension `int_dim` to dimension `1`.
-- `nn_h`: a differentiable function that must take vectors of dimension `in_dim + int_dim` to `out_dim`.
+# Examples
 
 ```jldoctest
-julia> in_dim, int_dim, out_dim = 3,6,5
-(3, 5, 5)
-
-julia> egnn = EEquivGraphConv(in_dim, int_dim, out_dim)
-EEquivGraphConv{Dense{typeof(identity), Matrix{Float32}, Vector{Float32}}, Dense{typeof(identity), Matrix{Float32}, Vector{Float32}}, Dense{typeof(identity), Matrix{Float32}, Vector{Float32}}}(Dense(8 => 5), Dense(5 => 1), Dense(8 => 5), 3, 5, 5)
-
-julia> m_len = 2*in_dim + 2
-8
-
-julia> nn_edge = Flux.Dense(m_len, int_dim)
-Dense(8 => 5)       # 45 parameters
+julia> in_dim, out_dim, pos_dim = 3, 5, 2
+(3, 5, 2)
 
-julia> nn_x = Flux.Dense(int_dim, 1)
-Dense(5 => 1)       # 6 parameters
-
-julia> nn_h = Flux.Dense(in_dim + int_dim, out_dim)
-Dense(8 => 5)       # 45 parameters
-
-julia> egnn = EEquivGraphConv(in_dim, nn_edge, nn_x, nn_h)
-EEquivGraphConv{Dense{typeof(identity), Matrix{Float32}, Vector{Float32}}, Dense{typeof(identity), Matrix{Float32}, Vector{Float32}}, Dense{typeof(identity), Matrix{Float32}, Vector{Float32}}}(Dense(8 => 5), Dense(5 => 1), Dense(8 => 5), 3, 5, 5)
+julia> egnn = EEquivGraphConv(in_dim=>out_dim, pos_dim, in_dim)
+EEquivGraphConv(ϕ_edge=Dense(10 => 5), ϕ_x=Dense(5 => 2), ϕ_h=Dense(8 => 5))
 ```
-"""
 
-struct EEquivGraphConv{E,X,H} <: MessagePassing
-   nn_edge::E
-   nn_x::X
-   nn_h::H
-   in_dim::Int
-   int_dim::Int
-   out_dim::Int
+See also [`WithGraph`](@ref) for training layer with static graph and [`EEquivGraphPE`](@ref) for positional encoding.
+"""
+struct EEquivGraphConv{X,E,H} <: AbstractGraphLayer
+    pe::X
+    nn_edge::E
+    nn_h::H
 end
 
 @functor EEquivGraphConv
 
-function EEquivGraphConv(in_dim::Int, int_dim::Int, out_dim::Int; init=glorot_uniform)
-    m_len = 2in_dim + 2
-    nn_edge = Flux.Dense(m_len, int_dim; init=init)
-    nn_x = Flux.Dense(int_dim, 1; init=init)
-    nn_h = Flux.Dense(in_dim + int_dim, out_dim; init=init)
-    return EEquivGraphConv(nn_edge, nn_x, nn_h, in_dim, int_dim, out_dim)
-end
+Flux.trainable(l::EEquivGraphConv) = (l.pe, l.nn_edge, l.nn_h)
 
-function EEquivGraphConv(in_dim::Int, nn_edge, nn_x, nn_h)
-    m_len = 2in_dim + 2
-    int_dim = Flux.outputsize(nn_edge, (m_len, 2))[1]
-    out_dim = Flux.outputsize(nn_h, (in_dim + int_dim, 2))[1]
-    return EEquivGraphConv(nn_edge, nn_x, nn_h, in_dim, int_dim, out_dim)
+function EEquivGraphConv(ch::Pair{Int,Int}, pos_dim::Int, edge_dim::Int; init=glorot_uniform)
+    in_dim, out_dim = ch
+    nn_edge = Flux.Dense(2in_dim + edge_dim + 1, out_dim; init=init)
+    pe = EEquivGraphPE(out_dim=>pos_dim; init=init)
+    nn_h = Flux.Dense(in_dim + out_dim, out_dim; init=init)
+    return EEquivGraphConv(pe, nn_edge, nn_h)
 end
 
-function ϕ_edge(egnn::EEquivGraphConv, h_i, h_j, dist, a)
-    N = size(h_i, 2)
-    return egnn.nn_edge(vcat(h_i, h_j, dist, ones(N)' * a))
-end
-
-ϕ_x(egnn::EEquivGraphConv, m_ij) = egnn.nn_x(m_ij)
-
-function message(egnn::EEquivGraphConv, v_i, v_j, e)
-    in_dim = egnn.in_dim
-    h_i = v_i[1:in_dim,:]
-    h_j = v_j[1:in_dim,:]
+ϕ_edge(l::EEquivGraphConv, h_i, h_j, dist, a) = l.nn_edge(vcat(h_i, h_j, dist, a))
 
-    N = size(h_i, 2)
+function message(l::EEquivGraphConv, h_i, h_j, x_i, x_j, e)
+    dist = sum(abs2, x_i - x_j; dims=1)
+    return ϕ_edge(l, h_i, h_j, dist, e)
+end
 
-    x_i = v_i[in_dim+1:end,:]
-    x_j = v_j[in_dim+1:end,:]
+update(l::EEquivGraphConv, m, h) = l.nn_h(vcat(h, m))
 
-    if isnothing(e)
-        a = 1
-    else
-        a = e[1]
-    end
+# For variable graph
+function(egnn::EEquivGraphConv)(fg::AbstractFeaturedGraph)
+    nf = node_feature(fg)
+    ef = edge_feature(fg)
+    pf = positional_feature(fg)
+    GraphSignals.check_num_nodes(fg, nf)
+    GraphSignals.check_num_edges(fg, ef)
+    _, V, X = propagate(egnn, graph(fg), ef, nf, pf, +)
+    return ConcreteFeaturedGraph(fg, nf=V, pf=X)
+end
 
-    dist = sum(abs2.(x_i - x_j); dims=1)
-    edge_msg = ϕ_edge(egnn, h_i, h_j, dist, a)
-    output_vec = vcat(edge_msg, (x_i - x_j) .* ϕ_x(egnn, edge_msg)[1], ones(N)')
-    return reshape(output_vec, :, N)
+# For static graph
+function(l::EEquivGraphConv)(el::NamedTuple, H::AbstractArray, E::AbstractArray, X::AbstractArray)
+    GraphSignals.check_num_nodes(el.N, H)
+    GraphSignals.check_num_nodes(el.N, X)
+    GraphSignals.check_num_edges(el.E, E)
+    _, V, X = propagate(l, el, E, H, X, +)
+    return V, X
 end
 
-function update(e::EEquivGraphConv, m, h)
-    N = size(m, 2)
-    mi = m[1:e.int_dim,:]
-    x_msg = m[e.int_dim+1:end-1,:]
-    M = m[end,:]
+function Base.show(io::IO, l::EEquivGraphConv)
+    print(io, "EEquivGraphConv(ϕ_edge=", l.nn_edge)
+    print(io, ", ϕ_x=", l.pe.nn)
+    print(io, ", ϕ_h=", l.nn_h)
+    print(io, ")")
+end
 
-    C = 1 ./ (M.-1)
-    C = reshape(C, :, N)
+function aggregate_neighbors(::EEquivGraphConv, el::NamedTuple, aggr, E)
+    batch_size = size(E)[end]
+    dstsize = (size(E, 1), el.N, batch_size)
+    xs = batched_index(el.xs, batch_size)
+    return _scatter(aggr, E, xs, dstsize)
+end
 
-    nn_node_out = e.nn_h(vcat(h[1:e.in_dim,:], mi))
+aggregate_neighbors(::EEquivGraphConv, el::NamedTuple, aggr, E::AbstractMatrix) = _scatter(aggr, E, el.xs)
 
-    coord_dim = size(h,1) - e.in_dim
+@inline aggregate_neighbors(::EEquivGraphConv, ::NamedTuple, ::Nothing, E) = nothing
+@inline aggregate_neighbors(::EEquivGraphConv, ::NamedTuple, ::Nothing, ::AbstractMatrix) = nothing
 
-    z = zeros(e.out_dim + coord_dim, N)
-    z[1:e.out_dim,:] = nn_node_out
-    z[e.out_dim+1:end,:] = h[e.in_dim+1:end,:] + C .* x_msg
-    return z
+function propagate(l::EEquivGraphConv, sg::SparseGraph, E, V, X, aggr)
+    el = to_namedtuple(sg)
+    return propagate(l, el, E, V, X, aggr)
 end
 
-function(egnn::EEquivGraphConv)(fg::AbstractFeaturedGraph)
-    X = node_feature(fg)
-    GraphSignals.check_num_nodes(fg, X)
-    _, V, _ = propagate(egnn, graph(fg), nothing, X, nothing, +, nothing, nothing)
-    return ConcreteFeaturedGraph(fg, nf=V)
+function propagate(l::EEquivGraphConv, el::NamedTuple, E, V, X, aggr)
+    E = message(
+        l, _gather(V, el.xs), _gather(V, el.nbrs),
+        _gather(X, el.xs), _gather(X, el.nbrs),
+        _gather(E, el.es)
+        )
+    X = positional_encode(l.pe, el, X, E)
+    Ē = aggregate_neighbors(l, el, aggr, E)
+    V = update(l, Ē, V)
+    return E, V, X
 end
+
+WithGraph(fg::AbstractFeaturedGraph, l::EEquivGraphConv) = WithGraph(to_namedtuple(fg), l)
+(wg::WithGraph{<:EEquivGraphConv})(args...) = wg.layer(wg.graph, args...)