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Parallelizable Riemannian Alternating Direction Method of Multipliers for Non-convex Pose Graph Optimization

Xin Chen, Chunfeng Cui, Deren Han, Liqun Qi

TL;DR

The paper tackles the scalability of pose graph optimization (PGO) in SLAM by introducing PRADMM, a parallelizable, Riemannian ADMM framework. It reformulates PGO via variable duplication and structured splitting, yielding per-vertex updates that are entirely parallelizable and admit closed-form solutions, with an overall complexity of $\mathcal{O}(m/n)$. Global convergence is established under mild conditions, and the method supports an extended dual-step range $\tau\in(0,2)$. Empirically, PRADMM outperforms existing graph-SLAM methods in speed while preserving accuracy across synthetic and real SLAM benchmarks, demonstrating strong scalability to large graph sizes.

Abstract

Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval $(0,2)$. Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.

Parallelizable Riemannian Alternating Direction Method of Multipliers for Non-convex Pose Graph Optimization

TL;DR

The paper tackles the scalability of pose graph optimization (PGO) in SLAM by introducing PRADMM, a parallelizable, Riemannian ADMM framework. It reformulates PGO via variable duplication and structured splitting, yielding per-vertex updates that are entirely parallelizable and admit closed-form solutions, with an overall complexity of . Global convergence is established under mild conditions, and the method supports an extended dual-step range . Empirically, PRADMM outperforms existing graph-SLAM methods in speed while preserving accuracy across synthetic and real SLAM benchmarks, demonstrating strong scalability to large graph sizes.

Abstract

Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval . Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.
Paper Structure (43 sections, 11 theorems, 124 equations, 6 figures, 8 tables, 2 algorithms)

This paper contains 43 sections, 11 theorems, 124 equations, 6 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Pose Graph Optimization
  4. Riemannian ADMM
  5. Main Contributions
  6. Model
  7. Algorithm
  8. Reformulation via Structured Splitting
  9. PRADMM
  10. Subproblems
  11. $\tilde{\bm{p}}$--subproblem:
  12. $\tilde{q}_{i}$, $\bm{t}_{i}$, and $\bm{s}_{i}$ subproblems
  13. Convergence Analysis
  14. Subsequential Convergence
  15. Global Convergence
  16. ...and 28 more sections

Key Result

Lemma 1

adachi2017solving Consider the spherical constrained problem where $A \in \mathbb{S}^{n \times n}$. For a solution $(x^{*},\lambda^{*})$ of the problem eq-lemma4.2-1, the multiplier $\lambda^{*}$ is equal to the largest real eigenvalue of $\tilde{Q}(\lambda)$, where

Figures (6)

  • Figure 1: A pose-graph representation of the SLAM process.
  • Figure 2: The comparison of cube trajectory with $\hat{n}=7$. From left to right are the real trajectory, the corrupted trajectory, and the recovered results by mG-N, SE-sync, RS+PS, PieADMM, and PRADMM, respectively.
  • Figure 3: The trend of CPU time along with different $\hat{n}$ for cube datasets. Left: $p_{cube}=0.3$. Right: $p_{cube}=0.9$.
  • Figure 4: The comparison of cube trajectory with $\hat{n}=5$. From left to right are the real trajectory, the corrupted trajectory and the recovered results by mG-N, SE-sync, RS+PS, PieADMM and PRADMM, respectively.
  • Figure 5: The trend of the number of edges, relative error, and CPU time along with different $\hat{n}$ under $\sigma_{t}^{rel}=0.1$, $\sigma_{r}=0.1$ and $p_{cube}\in\{0.3,0.6,0.9\}$.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Theorem 2
  • Remark 2
  • Lemma 4
  • Definition 1
  • Lemma 5
  • ...and 13 more