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Certifiable Estimation with Factor Graphs

Zhexin Xu, Nikolas R. Sanderson, Hanna Jiamei Zhang, David M. Rosen

TL;DR

Structural preservation in factor graph structure is preserved under Shor's relaxation and Burer-Monteiro factorization, enabling the Riemannian Staircase methodology for certifiable estimation to be implemented using the same mature, highly-performant factor graph libraries and workflows already ubiquitously employed throughout robotics and computer vision.

Abstract

Factor graphs provide a convenient modular modeling language that enables practitioners to design and deploy high-performance robotic state estimation systems by composing simple, reusable building blocks. However, inference in these models is typically performed using local optimization methods that can converge to suboptimal solutions, a serious reliability concern in safety-critical applications. Conversely, certifiable estimators based on convex relaxation can recover verifiably globally optimal solutions in many practical settings, but the computational cost of solving their large-scale relaxations necessitates specialized, structure-exploiting solvers that require substantial expertise to implement, significantly hampering practical deployment. In this paper, we show that these two paradigms, which have thus far been treated as independent in the literature, can be naturally synthesized into a unified framework for certifiable factor graph optimization. The key insight is that factor graph structure is preserved under Shor's relaxation and Burer-Monteiro factorization: applying these transformations to a QCQP with an associated factor graph representation yields a lifted problem admitting a factor graph model with identical connectivity, in which variables and factors are simple one-to-one algebraic transformations of those in the original QCQP. This structural preservation enables the Riemannian Staircase methodology for certifiable estimation to be implemented using the same mature, highly-performant factor graph libraries and workflows already ubiquitously employed throughout robotics and computer vision, making certifiable estimation as straightforward to design and deploy as conventional factor graph inference.

Certifiable Estimation with Factor Graphs

TL;DR

Structural preservation in factor graph structure is preserved under Shor's relaxation and Burer-Monteiro factorization, enabling the Riemannian Staircase methodology for certifiable estimation to be implemented using the same mature, highly-performant factor graph libraries and workflows already ubiquitously employed throughout robotics and computer vision.

Abstract

Factor graphs provide a convenient modular modeling language that enables practitioners to design and deploy high-performance robotic state estimation systems by composing simple, reusable building blocks. However, inference in these models is typically performed using local optimization methods that can converge to suboptimal solutions, a serious reliability concern in safety-critical applications. Conversely, certifiable estimators based on convex relaxation can recover verifiably globally optimal solutions in many practical settings, but the computational cost of solving their large-scale relaxations necessitates specialized, structure-exploiting solvers that require substantial expertise to implement, significantly hampering practical deployment. In this paper, we show that these two paradigms, which have thus far been treated as independent in the literature, can be naturally synthesized into a unified framework for certifiable factor graph optimization. The key insight is that factor graph structure is preserved under Shor's relaxation and Burer-Monteiro factorization: applying these transformations to a QCQP with an associated factor graph representation yields a lifted problem admitting a factor graph model with identical connectivity, in which variables and factors are simple one-to-one algebraic transformations of those in the original QCQP. This structural preservation enables the Riemannian Staircase methodology for certifiable estimation to be implemented using the same mature, highly-performant factor graph libraries and workflows already ubiquitously employed throughout robotics and computer vision, making certifiable estimation as straightforward to design and deploy as conventional factor graph inference.
Paper Structure (47 sections, 1 theorem, 61 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 47 sections, 1 theorem, 61 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Related work
  4. Factor graphs for robot perception
  5. Modeling perception problems with factor graphs
  6. Maximum likelihood estimation in factor graphs
  7. Certifiable estimation
  8. Shor's relaxation
  9. Solving the semidefinite relaxation
  10. Exploiting low-rank structure via Burer-Monteiro factorization
  11. Verifying global optimality
  12. Riemannian Staircase
  13. Certifiable factor graph optimization
  14. Factor graph structure in QCQPs
  15. Factor graph structure implies block sparsity and separability in QCQP data matrices
  16. ...and 32 more sections

Key Result

Theorem 1

Suppose that Shor's relaxation Shor_relaxation satisfies Slater's constraint qualification, and let $Y \in \mathbb{R}^{r \times p}$ be a KKT point of the BM factorization rank_p_BM_factorization (with corresponding Lagrange multiplier $\lambda \in \mathbb{R}^{M}$) that satisfies the linear independe

Figures (3)

  • Figure 1: An example factor graph. This graph models a simple landmark SLAM problem containing $N$ variables $X_n$ and $K$ factors $p_k$. The variables consist of both robot poses (dark gray) and landmarks (light gray), while the factors consist of a unary prior$p_1$ on the initial robot pose $X_1$, binary odometry factors between successive robot poses (blue), landmark observations (yellow), and loop closure detections (red).
  • Figure 2: Overview of our framework for certifiable factor graph optimization. Our approach accepts as input a factor graph model of a QCQP-representable estimation task, and constructs its corresponding rank-$p$ BM-factored Shor relaxation (left). This BM-factored SDP naturally preserves the original QCQP's factor graph structure, as reflected in the block sparsity pattern of the objective matrix $Q$ and the separable block-diagonal structure of the constraint matrices $A_m$. Consequently, the constraints on each block variable $Y_n$ in the resulting BM-factored SDP determine a feasible set $\mathcal{M}^{(p)}$ with the structure of a Cartesian product; this enables transformation of the (constrained) BM-factored SDP to an unconstrained intrinsic form with two key benefits: (1) automatic problem formulation directly from the (lifted) factor graph structure, and (2) efficient local optimization via smooth manifold-based optimization methods. Following the Riemannian staircase procedure (top), solutions are recovered via a sequence of local optimizations over lifted factor graphs, interleaved with certification steps. For constructing the optimality certificate $S$ (right), the block-separable structure of the constraints enables decomposition into $N$independent computations that can be solved efficiently in parallel.
  • Figure 3: Globally optimal results on select SLAM benchmark datasets encompassing pose-graph optimization, landmark, and range-aided problems. The blue denotes the trajectory that the robot traversed, while the red denotes the landmarks utilized in the case of landmark and range-aided SLAM.

Theorems & Definitions (3)

  • Theorem 1: Theorem 4 of rosen2020Scalable
  • Remark 1: Feasibility in local optimization in the Riemannian Staircase
  • Remark 2: Range-aided SLAM tolerances