The content introduces two connection-Laplacian-based problems on graphs:
Graph Tikhonov smoothing: Smoothing a complex-valued signal on a graph by minimizing a quadratic objective that penalizes functions incoherent with the connection.
Angular synchronization: Recovering a set of unknown angles from noisy pairwise offset measurements, formulated as an optimization problem on a graph.
The authors propose two types of RandNLA estimators to solve these problems:
A Feynman-Kac formula-based local estimator, which propagates values along random walks on the graph.
A global estimator that propagates values along the branches of sampled Multi-Type Spanning Forests (MTSFs) - graph decompositions into rooted trees and unicycles.
The MTSF-based estimator is the main contribution, and is shown to be unbiased and have an expected runtime linear in the number of edges. The authors also propose variance reduction techniques to improve the practical performance of their estimators.
Numerical experiments demonstrate that the MTSF-based estimators can outperform standard deterministic solvers, especially on dense graphs, by providing significant speedups for equivalent precision.
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