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Efficient Randomized Algorithms for Tikhonov Smoothing and Angular Synchronization on Graphs with Connections


Core Concepts
The authors propose novel randomized numerical linear algebra (RandNLA) estimators for solving connection-Laplacian-based problems on graphs, including Tikhonov smoothing and angular synchronization. Their estimators leverage propagations along branches of Multi-Type Spanning Forests (MTSFs), a graph decomposition that captures both the topology and the connection-induced rotations.
Abstract
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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Deeper Inquiries

How can the proposed RandNLA estimators be extended to handle more general types of connections beyond 2D rotations, such as 3D rotations encountered in applications like cryo-EM

To extend the proposed RandNLA estimators to handle more general types of connections beyond 2D rotations, such as 3D rotations encountered in applications like cryo-EM, we can modify the connection maps to accommodate the additional dimensions. Instead of unitary complex numbers representing 2D rotations, we can use unitary matrices representing 3D rotations. Each edge in the graph would then be associated with a 3x3 unitary matrix encoding the rotation in 3D space. The connection Laplacian would be defined based on these 3D rotation matrices, and the propagation of values along the edges would involve matrix multiplication instead of complex number multiplication. By adapting the algorithms to work with these 3D rotations, we can handle more complex connections in applications like cryo-EM where 3D rotations are prevalent.

What are the theoretical limits of the proposed MTSF-based estimators in terms of convergence rates and how do they compare to deterministic solvers

The theoretical limits of the proposed MTSF-based estimators in terms of convergence rates depend on the specific characteristics of the graph, such as its topology, density, and the distribution of the connection angles. In general, MTSF-based estimators offer advantages in terms of computational efficiency and scalability compared to deterministic solvers, especially for large graphs where exact solutions are computationally prohibitive. However, the convergence rates of the estimators may be slower than those of deterministic solvers, particularly in cases where the graph is very sparse or the connections introduce significant incoherence. The trade-off lies in the ability to provide approximate solutions efficiently, with the potential for variance reduction techniques to improve the accuracy of the estimations. Comparing the convergence rates of MTSF-based estimators with deterministic solvers would involve analyzing the spectral properties of the connection Laplacian and the impact of the graph structure on the convergence behavior.

Could the ideas behind the MTSF-based estimators be applied to other graph-based problems beyond smoothing and synchronization, such as graph sparsification or spectral clustering

The ideas behind the MTSF-based estimators can indeed be applied to other graph-based problems beyond smoothing and synchronization. For example, in graph sparsification, the concept of Multi-Type Spanning Forests can be utilized to efficiently sample subsets of edges that preserve the essential connectivity of the graph while reducing its size. By leveraging the random decomposition of the graph into MTSFs, one can identify key edges that maintain the graph's structure and connectivity. Similarly, in spectral clustering, MTSF-based estimators can be used to identify clusters in the graph by propagating information along the branches of the MTSFs to uncover the underlying community structure. The flexibility and scalability of MTSF-based estimators make them versatile tools for a wide range of graph-based problems beyond the specific applications of smoothing and synchronization.
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