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

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 $Δ$, 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 à 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 $Δ+ q \mathbb{I}$ with $q>0$ -- on sparsification guarantees.

Figures (17)

• Figure 1: Different spanning subgraphs of a grid graph. In blue, we show the root nodes used for sampling the subgraphs with the cycle-popping algorithm of \ref{['sec:sampling_a_multitype_spanning_forest']}, as well as the cycles in which some of the trees are rooted.
• Figure 2: Different oriented spanning subgraphs of a grid graph: an oriented Cycle-Rooted Spanning Forest (CRSF), an oriented Multi-Type Spanning Forest (MTSF) and an oriented Spanning Forest (SF). The root of an oriented tree (in blue) is the node with no out-going edge.
• Figure 3: Sparsify-and-eigensolve the magnetic Laplacian of MUN and ERO graphs. On the top row, we display the distance between the top eigenvectors of the magnetic Laplacian and its sparsifier. On the bottom row, we report the ranking recovery with Sync-Rank based on the sparsifier, in terms of Kendall's tau distance to the planted ranking.
• Figure 4: Sparsify-and-precondition the magnetic Laplacian of a MUN$(n,p,\eta)$ graph with $n= 2000$ and $p=0.01$. We display $\mathop{\mathrm{\mathrm{cond}}}\nolimits(\widetilde{\Delta} ^{-1}\Delta)$. Results are averaged over $3$ independent estimates $\widetilde{\Delta}$ for a fixed MUN graph.
• Figure 5: Sparsify-and-precondition the magnetic Laplacian of ERO$(n,p,\eta)$ graph with $n= 2000$ and $p=0.01$. We display $\mathop{\mathrm{\mathrm{cond}}}\nolimits(\widetilde{\Delta} ^{-1}\Delta)$. Results are averaged over $3$ independent estimates $\widetilde{\Delta}$ for a fixed ERO graph.

Theorems & Definitions (29)

• Definition 1: Discrete DPP
• Proposition 1: $\mathrm{L}$-ensemble
• Remark 1: Regularized Laplacian.
• Remark 2: $q = 0$
• Proposition 2: Sparsification from one MTSF
• proof
• Proposition 3: Sparse Cholesky decomposition for the Laplacian of one MTSF
• Theorem 1: Sparsification with a batch of MTSFs
• proof
• Theorem 2: Chernoff bound with intrinsic dimension