Sparsification of the regularized magnetic Laplacian with multi-type spanning forests

TL;DR

This paper introduces a novel method for sparsifying magnetic Laplacians in large graphs using multi-type spanning forests sampled via determinantal point processes, with applications in ranking and semi-supervised learning.

Contribution

It proposes a new sparsification technique for magnetic Laplacians based on multi-type spanning forests and provides statistical guarantees for the estimators involved.

Findings

Effective spectral approximation of magnetic Laplacians with few edges.

Sampling method captures angular inconsistencies in connection graphs.

Applications demonstrated in ranking and semi-supervised learning.

Abstract

In this paper, we consider a ${\rm U}(1)$-connection graph, that is, a graph where each oriented edge is endowed with a unit modulus complex number that is conjugated under orientation flip. A natural replacement for the combinatorial Laplacian is then the magnetic Laplacian, an Hermitian matrix that includes information about the graph's connection. Magnetic Laplacians appear, e.g., in the problem of angular synchronization. In the context of large and dense graphs, we study here sparsifiers of the magnetic Laplacian $\Delta$, i.e., spectral approximations based on subgraphs with few edges. Our approach relies on sampling multi-type spanning forests (MTSFs) using a custom determinantal point process, a probability distribution over edges that favours diversity. In a word, an MTSF is a spanning subgraph whose connected components are either trees or cycle-rooted trees. The latter partially capture the angular inconsistencies of the connection graph, and thus provide a way to compress the information contained in the connection. Interestingly, when the connection graph has weakly inconsistent cycles, samples from the determinantal point process under consideration can be obtained \`a la Wilson, using a random walk with cycle popping. We provide statistical guarantees for a choice of natural estimators of the connection Laplacian, and investigate two practical applications of our sparsifiers: ranking with angular synchronization and graph-based semi-supervised learning. From a statistical perspective, a side result of this paper of independent interest is a matrix Chernoff bound with intrinsic dimension, which allows considering the influence of a regularization -- of the form $\Delta + q \mathbb{I}$ with $q>0$ -- on sparsification guarantees.