LASE: Learned Adjacency Spectral Embeddings

TL;DR

LASE is a neural network architecture that learns spectral embeddings of graph nodes through gradient descent unrolling, combining GCN and GAT modules, and outperforms traditional eigendecomposition in graph tasks.

Contribution

This work introduces a novel neural architecture called LASE that learns spectral embeddings via algorithm unrolling, offering interpretability, efficiency, and improved performance in graph tasks.

Findings

LASE outperforms eigendecomposition routines in approximation error.

LASE achieves superior results in link prediction and node classification.

The architecture is robust to unobserved edges and integrates seamlessly into larger models.

Abstract

We put forth a principled design of a neural architecture to learn nodal Adjacency Spectral Embeddings (ASE) from graph inputs. By bringing to bear the gradient descent (GD) method and leveraging the principle of algorithm unrolling, we truncate and re-interpret each GD iteration as a layer in a graph neural network (GNN) that is trained to approximate the ASE. Accordingly, we call the resulting embeddings and our parametric model Learned ASE (LASE), which is interpretable, parameter efficient, robust to inputs with unobserved edges, and offers controllable complexity during inference. LASE layers combine Graph Convolutional Network (GCN) and fully-connected Graph Attention Network (GAT) modules, which is intuitively pleasing since GCN-based local aggregations alone are insufficient to express the sought graph eigenvectors. We propose several refinements to the unrolled LASE architecture (such as sparse attention in the GAT module and decoupled layerwise parameters) that offer favorable approximation error versus computation tradeoffs; even outperforming heavily-optimized eigendecomposition routines from scientific computing libraries. Because LASE is a differentiable function with respect to its parameters as well as its graph input, we can seamlessly integrate it as a trainable module within a larger (semi-)supervised graph representation learning pipeline. The resulting end-to-end system effectively learns ``discriminative ASEs'' that exhibit competitive performance in supervised link prediction and node classification tasks, outperforming a GNN even when the latter is endowed with open loop, meaning task-agnostic, precomputed spectral positional encodings.