Practical and Performant Enhancements for Maximization of Algebraic Connectivity

TL;DR

This work tackles scalable state estimation on ever-growing graphs by improving the Maximizing Algebraic Connectivity (MAC) sparsification framework. It introduces a fast Fiedler-value solver based on shift-invert Krylov methods, evaluates various Frank–Wolfe line-search strategies, and eliminates the need for a user-specified backbone by providing connectivity-guaranteeing rounding and a spectrally informed backbone heuristic. The proposed methods yield substantial runtime reductions (often about a 2x speedup) and higher algebraic connectivity $\lambda_2$ while ensuring connected sparsified graphs, enabling real-time estimation on large pose graphs. These advances enhance both the practicality and reliability of graph-based long-term perception and estimation in robotics, with direct impact on SLAM, PGO, and structure-from-motion pipelines.

Abstract

Long-term state estimation over graphs remains challenging as current graph estimation methods scale poorly on large, long-term graphs. To address this, our work advances a current state-of-the-art graph sparsification algorithm, maximizing algebraic connectivity (MAC). MAC is a sparsification method that preserves estimation performance by maximizing the algebraic connectivity, a spectral graph property that is directly connected to the estimation error. Unfortunately, MAC remains computationally prohibitive for online use and requires users to manually pre-specify a connectivity-preserving edge set. Our contributions close these gaps along three complementary fronts: we develop a specialized solver for algebraic connectivity that yields an average 2x runtime speedup; we investigate advanced step size strategies for MAC's optimization procedure to enhance both convergence speed and solution quality; and we propose automatic schemes that guarantee graph connectivity without requiring manual specification of edges. Together, these contributions make MAC more scalable, reliable, and suitable for real-time estimation applications.

Figures (5)

• Figure 1: We sparsify a graph by pruning edges in a way that maintains the graph’s structural properties, as quantified by its algebraic connectivity. (Top) Example of a "mock" graph undergoing sparsification. (Bottom) Solution of the estimation problem on the sparsified graph retains the overall shape of the original while using significantly fewer edges.
• Figure 2: Overview of MAC algorithm with highlighted contributions: MAC doherty2024mac is given a graph and prespecified "backbone" (fixed edge set), then applies a Frank–Wolfe procedure with a decaying step size, and finally rounds the solution to a binary one. To reduce total runtime and broaden MAC’s applicability to problems without a pre-specified backbone, we investigate three main questions: 1. Can we increase MAC's runtime through Faster Fiedler value calculations (\ref{['sec:FiedlerValue']})?; 2. Can we increase MAC's convergence rate through intelligent step size and linesearch routines (\ref{['sec:lineSearch']})?; 3. Can we extend MAC to problems without a given fixed edge set through rounding functions $\Pi_K$(\ref{['sec:rounding']}) and automatic backbone edge specification (\ref{['sec:effR']})?
• Figure 3: TIERSyu2023fusing dataset sparsified to 80% edges via Frank-Wolfe. Notably, each iteration takes significantly less time when using our algorithm due to the faster Fiedler Pair solves (top), resulting in higher algebraic connectivity $\lambda_2$ in shorter time (bottom)
• Figure 4: Algebraic Connectivity vs Iteration: Each graph corresponds to the runs in \ref{['tab:linesearchTimes']}: As seen, the Exact, Backtracking, and PFW-Exact variants stall in optimization progress in a few iterates. While the Naive step size strategy initially takes a poor step, it does not suffer from this stalling behavior.
• Figure 5: Algebraic Connectivity vs Edge Budget. Graphs of $\lambda_2$ vs % edges kept. Shaded error bars indicate the one Std Dev. bound around the mean cost for Madow rounding strategies. Our heuristic backbone generally outperforms all other strategies, while interestingly the backbone-free approaches (MST, MSTMadow) generally perform the worst.