MAC: Graph Sparsification by Maximizing Algebraic Connectivity

TL;DR

This work addresses lifelong SLAM under memory and computation limits by sparsifying the measurement graph while preserving estimator quality. It introduces MAC, a first-order method that maximizes the algebraic connectivity $\lambda_2(L(x))$ of the graph Laplacian under a fixed edge budget, via a convex relaxation solved with Frank-Wolfe and supergradients, followed by rounding. The approach yields formal post-hoc guarantees and scales to large SLAM problems, outperforming baselines in connectivity and SLAM accuracy on benchmark datasets and real multi-session data, often with orders of magnitude faster computation than SDP-based alternatives. The contribution includes a robust, fast sparsification pipeline with practical edge-rounding strategies (notably Madow sampling) and open-source code, enabling reliable, scalable graph design for SLAM and related robotics problems.

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.

Figures (6)

• Figure 1: Qualitative results for 3D pose-graph sparsification. Pose-graph optimization results for the three 3D benchmark datasets after pruning all but 20% of the original loop closures using each method: (Top) Grid, (Middle) Torus, (Bottom) Sphere.
• Figure 2: Quantitative results for 2D pose-graph sparsification. Pose-graph optimization results for (a) the Intel dataset, (b) the AIS2Klinik dataset, and (c) the City10K dataset with varying degrees of sparsity (as percent of candidate edges added). Left to right: The algebraic connectivity of the graphs obtained using each method (larger is better) with the shaded regions indicating the suboptimality gap for each MAC rounding procedure, the mean translation error and relative rotation error compared to a maximum-likelihood estimate computed for the full graph, i.e. with all edges retained (smaller is better; note the logarithmic scale), and the computation time (logarithmic scale) for each approach. For 100% loop closures, both algorithms return immediately, so no computation time is reported.
• Figure 3: Quantitative results for 3D pose-graph sparsification. Pose-graph optimization results for (a) the Grid dataset, (b) the Torus dataset, and (c) the Sphere dataset with varying degrees of sparsity (as percent of candidate edges added). Left to right: The algebraic connectivity of the graphs obtained using each method (larger is better) with the shaded regions indicating the suboptimality gap for each MAC rounding procedure, the mean translation error and relative rotation error compared to a maximum-likelihood estimate computed for the full graph, i.e. with all edges retained (smaller is better; note the logarithmic scale), and the computation time (logarithmic scale) for each approach. For 100% loop closures, both algorithms return immediately, so no computation time is reported.
• Figure 4: Point cloud reconstructions results for the NCLT Dataset. Top-down view of the LiDAR point clouds obtained from SLAM solutions computed using graphs sparsified by each approach to 20% of the total loop closures in the dataset. Point clouds are colored by height in the $z$-axis from blue to red.
• Figure 5: Qualitative results. Offset trajectory visualization comparing the results of the naïve baseline approach with MAC (Madow). Separate trajectories are distributed on the $z$-axis and ordered temporally (i.e. with the trajectory from the first session being the lowest and proceeding upward). Odometry edges are displayed in blue, while loop closures are shown in green. Different viewpoints are used to highlight the impact of poor connectivity in the case of the naïve method. While the first and second sessions are relatively well-connected throughout, the third session is poorly anchored relative to the first two. In contrast, the solution obtained by MAC (Madow) results in a qualitatively more even distribution of loop closure edges throughout, and in turn a higher-quality SLAM solution.

Theorems & Definitions (6)

• Lemma 1
• Theorem 2: Supergradients of $\lambda_2(L(x))$
• Theorem 3: A closed-form solution to Problem \ref{['prob:dir-subproblem']}
• Theorem 4: Theorem 3.5 of overton1993optimality
• proof : Proof of Theorem \ref{['thm:supergradient']}
• proof : Proof of Theorem \ref{['thm:dir-subproblem']}