SplineConv: Smooth Continuous Filters on Graphs

What SplineConv does

SplineConv generalizes standard image convolution to irregular graph structures:

1. Interpret edge attributes as pseudo-coordinates in a continuous d-dimensional space
2. Evaluate B-spline basis functions at these coordinates to get smooth filter weights
3. Apply the position-dependent filter to neighbor features and aggregate

On a regular grid (image), SplineConv recovers standard 2D convolution. On irregular meshes, it adapts to the local geometry.

PyG implementation

import torch
import torch.nn.functional as F
from torch_geometric.nn import SplineConv

class SplineNet(torch.nn.Module):
    def __init__(self, in_channels, hidden_channels, out_channels):
        super().__init__()
        self.conv1 = SplineConv(in_channels, hidden_channels,
                                 dim=2, kernel_size=5, degree=1)
        self.conv2 = SplineConv(hidden_channels, out_channels,
                                 dim=2, kernel_size=5, degree=1)

    def forward(self, x, edge_index, edge_attr):
        # edge_attr: pseudo-coordinates in [0, 1]^dim
        x = F.relu(self.conv1(x, edge_index, edge_attr))
        x = self.conv2(x, edge_index, edge_attr)
        return x

# For mesh data: edge_attr encodes relative positions
# For images: edge_attr encodes pixel offsets (normalized to [0,1])
model = SplineNet(in_channels=3, hidden_channels=64, out_channels=10)

When to use SplineConv

• Mesh processing. 3D mesh classification and segmentation where edge attributes encode geometric relationships (relative positions, normals, angles).
• Manifold learning. Data on surfaces or low-dimensional manifolds where continuous spatial filters are natural.
• Image-as-graph tasks. MNIST/CIFAR on superpixel graphs where edge attributes encode spatial offsets between superpixels.

When not to use SplineConv

• Graphs without spatial edge attributes. SplineConv requires meaningful pseudo-coordinates. Social networks, citation graphs, and knowledge graphs lack these.
• Discrete edge types. B-splines are designed for continuous coordinates. For categorical edge types, use RGCNConv or GINEConv.