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Uncertainty Quantification of Set-Membership Estimation in Control and Perception: Revisiting the Minimum Enclosing Ellipsoid

Yukai Tang, Jean-Bernard Lasserre, Heng Yang

TL;DR

The paper addresses uncertainty quantification in set-membership estimation by computing a minimum enclosing ellipsoid (MEE) of the SME set using the moment-SOS hierarchy. To make the approach scalable, it introduces three computational enhancements—constraint pruning, generalized relaxed Chebyshev center (GRCC), and a quaternion-based treatment of SO(3) geometry—that together enable practical, convergent MEE estimation with certificates under convexity. The authors demonstrate the method on system identification and object pose estimation, showing tighter uncertainty bounds than traditional least-squares approaches and robust performance on long trajectories. This work broadens the applicability of SME in control and perception by providing tractable, verifiable outer-approximations that support sampling and uncertainty quantification in real-world problems.

Abstract

Set-membership estimation (SME) outputs a set estimator that guarantees to cover the groundtruth. Such sets are, however, defined by (many) abstract (and potentially nonconvex) constraints and therefore difficult to manipulate. We present tractable algorithms to compute simple and tight overapproximations of SME in the form of minimum enclosing ellipsoids (MEE). We first introduce the hierarchy of enclosing ellipsoids proposed by Nie and Demmel (2005), based on sums-of-squares relaxations, that asymptotically converge to the MEE of a basic semialgebraic set. This framework, however, struggles in modern control and perception problems due to computational challenges. We contribute three computational enhancements to make this framework practical, namely constraints pruning, generalized relaxed Chebyshev center, and handling non-Euclidean geometry. We showcase numerical examples on system identification and object pose estimation.

Uncertainty Quantification of Set-Membership Estimation in Control and Perception: Revisiting the Minimum Enclosing Ellipsoid

TL;DR

The paper addresses uncertainty quantification in set-membership estimation by computing a minimum enclosing ellipsoid (MEE) of the SME set using the moment-SOS hierarchy. To make the approach scalable, it introduces three computational enhancements—constraint pruning, generalized relaxed Chebyshev center (GRCC), and a quaternion-based treatment of SO(3) geometry—that together enable practical, convergent MEE estimation with certificates under convexity. The authors demonstrate the method on system identification and object pose estimation, showing tighter uncertainty bounds than traditional least-squares approaches and robust performance on long trajectories. This work broadens the applicability of SME in control and perception by providing tractable, verifiable outer-approximations that support sampling and uncertainty quantification in real-world problems.

Abstract

Set-membership estimation (SME) outputs a set estimator that guarantees to cover the groundtruth. Such sets are, however, defined by (many) abstract (and potentially nonconvex) constraints and therefore difficult to manipulate. We present tractable algorithms to compute simple and tight overapproximations of SME in the form of minimum enclosing ellipsoids (MEE). We first introduce the hierarchy of enclosing ellipsoids proposed by Nie and Demmel (2005), based on sums-of-squares relaxations, that asymptotically converge to the MEE of a basic semialgebraic set. This framework, however, struggles in modern control and perception problems due to computational challenges. We contribute three computational enhancements to make this framework practical, namely constraints pruning, generalized relaxed Chebyshev center, and handling non-Euclidean geometry. We showcase numerical examples on system identification and object pose estimation.
Paper Structure (32 sections, 16 theorems, 77 equations, 7 figures, 6 tables, 1 algorithm)

This paper contains 32 sections, 16 theorems, 77 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Minimum Enclosing Ellipsoid by SOS Relaxations
  3. Computational Enhancement
  4. Pruning Redundant Constraints
  5. Generalized Relaxed Chebyshev Center
  6. Handling the non-Euclidean Geometry of $\mathrm{SO}(3)\xspace$
  7. Experiments
  8. System Identification (Example \ref{['ex:sme-system-id']})
  9. Comparison with Least Squares Estimation
  10. Identification of Random Linear Systems
  11. Linear Systems that Are Hard to Learn
  12. Constraints Pruning in A Long Trajectory
  13. Object Pose Estimation (Example \ref{['ex:sme-object-pose']})
  14. Conclusions
  15. Non-Gaussian Measurement Noise from Neural Networks
  16. ...and 17 more sections

Key Result

theorem 1

Assume ${\cal S}$ is Archimedean,The definition of Archimedeanness is given in blekherman12siam-semidefinite. One can make the Archimedean condition trivially hold by adding a constraint $M_\theta - \theta^{\mathsf{T}} \theta \geq 0$ to eq:basic-semialgebraic-set, which is easy for Examples ex:sme-s where $g_0(\theta) := 1$, $\mathrm{SOS}[\theta]$ is the set of SOS polynomials in $\theta$, and $\m

Figures (7)

  • Figure 1: Experimental results on system identification (Example \ref{['ex:sme-system-id']}).
  • Figure 2: Experimental results on object pose estimation (Example \ref{['ex:sme-object-pose']}).
  • Figure A1: Perspective plots (top) and Chi-square Q-Q plots (bottom) of the noise vectors generated by neural network measurements in the LM-O dataset. From left to right, they are respectively for $1\%$, $5\%$, $10\%$, $20\%$, $40\%$, $100\%$ noise vectors. The right-most graph shows the perspective plot for a Gaussian distribution that can be used as comparison.
  • Figure A2: Analogy of Proposition \ref{['prop:miniball-rotation']} in ${ {\mathbb R}^{2} }$. The black line segment is a subset of $S^1$ (the dotted circle). The solid blue circle is the minimum enclosing ball of the black line segment in ${ {\mathbb R}^{2} }$. The center of the solid blue circle, projected onto $S^1$, is the center of the minimum enclosing geodesic ball on $S^1$.
  • Figure A3: The certificate of finite convergence for TV Screen Example.
  • ...and 2 more figures

Theorems & Definitions (24)

  • theorem 1: MEE Approximation by SOS Programming
  • theorem 2: Generalized Relaxed Chebyshev Center
  • theorem 3: Minimum Enclosing Ball for Rotations
  • proposition A1: Condition for SOS Polynomials lasserre2009moments
  • theorem A1: Putinar's Positivstellensatz putinar1993positive
  • theorem A2: Representing measure lasserre2009moments
  • proof
  • proposition A2: Certificate of MEE
  • proof
  • lemma 1: John's Theorem
  • ...and 14 more