# Memory-Efficient Aggregations

The MessagePassing interface of PyG relies on a gather-scatter scheme to aggregate messages from neighboring nodes. For example, consider the message passing layer

$\mathbf{x}^{\prime}_i = \sum_{j \in \mathcal{N}(i)} \textrm{MLP}(\mathbf{x}_j - \mathbf{x}_i),$

that can be implemented as:

from torch_geometric.nn import MessagePassing

x = ...           # Node features of shape [num_nodes, num_features]
edge_index = ...  # Edge indices of shape [2, num_edges]

class MyConv(MessagePassing):
def __init__(self):

def forward(self, x, edge_index):
return self.propagate(edge_index, x=x)

def message(self, x_i, x_j):
return MLP(x_j - x_i)


Under the hood, the MessagePassing implementation produces a code that looks as follows:

from torch_scatter import scatter

x = ...           # Node features of shape [num_nodes, num_features]
edge_index = ...  # Edge indices of shape [2, num_edges]

x_j = x[edge_index[0]]  # Source node features [num_edges, num_features]
x_i = x[edge_index[1]]  # Target node features [num_edges, num_features]

msg = MLP(x_j - x_i)  # Compute message for each edge

# Aggregate messages based on target node indices
out = scatter(msg, edge_index[1], dim=0, dim_size=x.size(0), reduce="add")


While the gather-scatter formulation generalizes to a lot of useful GNN implementations, it has the disadvantage of explicitely materalizing x_j and x_i, resulting in a high memory footprint on large and dense graphs.

Luckily, not all GNNs need to be implemented by explicitely materalizing x_j and/or x_i. In some cases, GNNs can also be implemented as a simple-sparse matrix multiplication. As a general rule of thumb, this holds true for GNNs that do not make use of the central node features x_i or multi-dimensional edge features when computing messages. For example, the GINConv layer

$\mathbf{x}^{\prime}_i = \textrm{MLP} \left( (1 + \epsilon) \cdot \mathbf{x}_i + \sum_{j \in \mathcal{N}(i)} \mathbf{x}_j \right),$

is equivalent to computing

$\mathbf{X}^{\prime} = \textrm{MLP} \left( (1 + \epsilon) \cdot \mathbf{X} + \mathbf{A}\mathbf{X} \right),$

where $$\mathbf{A}$$ denotes a sparse adjacency matrix of shape [num_nodes, num_nodes]. This formulation allows to leverage dedicated and fast sparse-matrix multiplication implementations.

In PyG >= 1.6.0, we officially introduce better support for sparse-matrix multiplication GNNs, resulting in a lower memory footprint and a faster execution time. As a result, we introduce the SparseTensor class (from the torch-sparse package), which implements fast forward and backward passes for sparse-matrix multiplication based on the “Design Principles for Sparse Matrix Multiplication on the GPU” paper.

Using the SparseTensor class is straightforward and similar to the way scipy treats sparse matrices:

from torch_sparse import SparseTensor

sparse_sizes=(num_nodes, num_nodes))
# value is optional and can be None

# Obtain different representations (COO, CSR, CSC):

out = adj.matmul(x)    # Sparse-dense matrix multiplication

# Creating SparseTensor instances:


Our MessagePassing interface can handle both torch.Tensor and SparseTensor as input for propagating messages. However, when holding a directed graph in SparseTensor, you need to make sure to input the transposed sparse matrix to propagate():

conv = GCNConv(16, 32)
out1 = conv(x, edge_index)
assert torch.allclose(out1, out2)

conv = GINConv(nn=Sequential(Linear(16, 32), ReLU(), Linear(32, 32)))
out1 = conv(x, edge_index)
assert torch.allclose(out1, out2)


To leverage sparse-matrix multiplications, the MessagePassing interface introduces the message_and_aggregate() function (which fuses the message() and aggregate() functions into a single computation step), which gets called whenever it is implemented and receives a SparseTensor as input for edge_index. With it, the GINConv layer can now be implemented as follows:

from torch_sparse import matmul

class GINConv(MessagePassing):
def __init__(self):

def forward(self, x, edge_index):
out = self.propagate(edge_index, x=x)
return MLP((1 + eps) x + out)

def message(self, x_j):
return x_j



Playing around with the new SparseTensor format is straightforward since all of our GNNs work with it out-of-the-box. To convert the edge_index format to the newly introduced SparseTensor format, you can make use of the torch_geometric.transforms.ToSparseTensor transform:

import torch
import torch.nn.functional as F

from torch_geometric.nn import GCNConv
import torch_geometric.transforms as T
from torch_geometric.datasets import Planetoid

dataset = Planetoid("Planetoid", name="Cora", transform=T.ToSparseTensor())
data = dataset[0]
>>> Data(adj_t=[2708, 2708, nnz=10556], x=[2708, 1433], y=[2708], ...)

class GNN(torch.nn.Module):
def __init__(self):
super().__init__()
self.conv1 = GCNConv(dataset.num_features, 16, cached=True)
self.conv2 = GCNConv(16, dataset.num_classes, cached=True)

x = F.relu(x)
return F.log_softmax(x, dim=1)

model = GNN()

def train(data):
model.train()
loss = F.nll_loss(out, data.y)
loss.backward()
optimizer.step()
return float(loss)

for epoch in range(1, 201):
loss = train(data)


All code remains the same as before, except for the data transform via T.ToSparseTensor(). As an additional advantage, MessagePassing implementations that utilize the SparseTensor class are deterministic on the GPU since aggregations no longer rely on atomic operations.

Notably, the GNN layer execution slightly changes in case GNNs incorporate single or multi-dimensional edge information edge_weight or edge_attr into their message passing formulation, respectively. In particular, it is now expected that these attributes are directly added as values to the SparseTensor object. Instead of calling the GNN as

conv = GMMConv(16, 32, dim=3)
out = conv(x, edge_index, edge_attr)


we now execute our GNN operator as

conv = GMMConv(16, 32, dim=3)

Since this feature is still experimental, some operations, e.g., graph pooling methods, may still require you to input the edge_index format. You can convert adj_t back to (edge_index, edge_attr) via:
row, col, edge_attr = adj_t.t().coo()

Please let us know what you think of SparseTensor, how we can improve it, and whenever you encounter any unexpected behaviour.