MAC: Graph Sparsification by Maximizing Algebraic Connectivity (2403.19879v2)

Abstract: Simultaneous localization and mapping (SLAM) is a critical capability in autonomous navigation, but memory and computational limits make long-term application of common SLAM techniques impractical; a robot must be able to determine what information should be retained and what can safely be forgotten. In graph-based SLAM, the number of edges (measurements) in a pose graph determines both the memory requirements of storing a robot's observations and the computational expense of algorithms deployed for performing state estimation using those observations, both of which can grow unbounded during long-term navigation. Motivated by these challenges, we propose a new general purpose approach to sparsify graphs in a manner that maximizes algebraic connectivity, a key spectral property of graphs which has been shown to control the estimation error of pose graph SLAM solutions. Our algorithm, MAC (for maximizing algebraic connectivity), is simple and computationally inexpensive, and admits formal post hoc performance guarantees on the quality of the solution that it provides. In application to the problem of pose-graph SLAM, we show on several benchmark datasets that our approach quickly produces high-quality sparsification results which retain the connectivity of the graph and, in turn, the quality of corresponding SLAM solutions.

• The paper introduces a convex relaxation method for graph sparsification by maximizing algebraic connectivity to improve SLAM state estimation.
• It employs the Frank-Wolfe algorithm for efficient edge selection, outperforming existing sparsification techniques.
• Experimental results show that MAC maintains high estimation accuracy while significantly reducing computational overhead in robotics applications.

Maximizing Algebraic Connectivity for Graph Sparsification via a Convex Relaxation Approach

Introduction

Graph-based methods play a crucial role in various applications across robotics and computer vision, notably including the field of Simultaneous Localization and Mapping (SLAM). A fundamental issue in SLAM is the management of computational resources, where a robot must judiciously decide which pieces of information are crucial for accurate state estimation and which can be safely discarded. This paper introduces a novel graph sparsification technique aimed at retaining the most informative subset of edges within a pose graph by maximizing its algebraic connectivity. Through a convex relaxation strategy, the proposed Maximizing Algebraic Connectivity (MAC) algorithm achieves this goal efficiently and with formal performance guarantees.

Graph Sparsification and Algebraic Connectivity

Graph sparsification seeks to reduce the complexity of a graph while preserving its essential characteristics. In the context of SLAM, this equates to selecting a subset of all available measurements (edges) that maintains the estimation quality. Central to this process is the concept of algebraic connectivity, a spectral property of the graph's Laplacian, which correlates directly with the graph's robustness and the accuracy of the estimation it supports.

The algebraic connectivity of a graph is defined as the second smallest eigenvalue of its Laplacian matrix. A higher value indicates a more robustly connected graph, which in turn implies a better-posed estimation problem in SLAM. The problem of maximizing this algebraic connectivity, subject to a constraint on the number of edges, forms the crux of the MAC algorithm.

The MAC Algorithm: A Spectral Approach

Given the NP-Hard nature of directly maximizing algebraic connectivity, MAC adopts a convex relaxation approach, transforming the original combinatorial problem into a tractable convex optimization problem. This relaxation involves allowing the solution space to include fractional edge selections, leading to a concave maximization problem over a convex set. Specifically, MAC maximizes a relaxation of the algebraic connectivity over the set of graphs formed by any convex combination of the candidate edges subject to a fixed budget.

The core of the MAC algorithm is the application of the Frank-Wolfe algorithm, a first-order optimization method that iteratively linearizes the objective function and solves a linear program over the convex set at each step. This method is particularly suitable due to its simplicity and efficiency in exploiting the problem structure, including the ability to leverage efficient eigenvalue computations and the inherent sparsity of the solution space.

Experimental Results and Implications

The efficacy of MAC is demonstrated through extensive experiments on both synthetic and real-world SLAM datasets. These experiments highlight MAC's capability to produce sparsified graphs that closely approximate the connectivity of the original graph, thereby enabling high-quality state estimation with reduced computational overhead. Notably, the proposed method outperforms existing sparsification techniques in terms of preserving algebraic connectivity and consequently the fidelity of the SLAM solution.

Practical applications of the MAC algorithm extend beyond SLAM, including sensor network optimization, communication-constrained multi-robot systems, and incremental sparsification for lifelong SLAM. Furthermore, the introduction of formal suboptimality guarantees provides a foundation for future theoretical exploration, particularly in understanding the relationship between the algebraic connectivity of the solution to the convex relaxation and the original combinatorial problem.

Conclusion

The Maximizing Algebraic Connectivity algorithm represents a significant advancement in graph sparsification for robotics applications, particularly in the domain of SLAM. By efficiently maximizing the algebraic connectivity of a graph, MAC ensures that essential information is retained for accurate state estimation while minimizing computational requirements. The approach's generality, coupled with its formal performance guarantees, establishes a solid groundwork for further research and practical implementation in resource-constrained robotic systems.

Future developments may explore adaptive and incremental sparsification strategies, alternative formulations to accommodate varying constraints, and extensions to other problem domains where graph structure influences system performance.