Laplacian Positional Encoding

The problem, from first principles

Laplacian Positional Encoding (LPE) uses eigenvectors of the graph Laplacian as node features. The k smallest eigenvalues/eigenvectors capture global graph structure, useful for node classification and graph tasks.

From theory to code

Implement:

laplacian_positional_encoding(adj, k=10)

Computes Laplacian PE for all nodes.

Constraints

• adj shape: (num_nodes, num_nodes).
• k: number of eigenvectors to keep.
• Return shape: (num_nodes, k).