Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Convergence of the Extended Kalman Filter on Stiefel Manifolds when Observing a Constant Particle with Measurement Errors

Jordi-Lluís Figueras, Aron Persson, Lauri Viitasaari

TL;DR

This work addresses manifold-valued filtering where observations lie on Stiefel manifolds while the latent state evolves in Euclidean space, and proves convergence guarantees for an extended Kalman filter on $\mathrm{St}_{n,k}(\mathbb{K})$ under a constant-process assumption. It develops a projection-based measurement model $Z_m=\mathrm{pr}(\mathrm{pr}(X_{t_m})+\varepsilon_m)$, introduces a projected-normal framework, and uses a Padé-based variance approximation to implement a practical EKF on manifolds. The main theoretical contribution shows that when the true process is constant ($A=0$, $\nu=0$), the EKF mean $\mu^K_m$ converges in probability to the true state $X_0$ and the Kalman variance $P^K_m$ vanishes; proofs rely on projection-mean commutativity and convergence analysis of Kalman gains. Numerical simulations on multiple Stiefel manifolds demonstrate convergence behavior and reveal the impact of measurement noise and dimensionality on convergence speed, supporting the method's practicality for directional and manifold-valued filtering.

Abstract

In this paper we first introduce the setting of filtering on Stiefel manifolds. Then, assuming the underlying system process is constant, the convergence of the extended Kalman filter with Stiefel manifold-valued observations is proved. This corresponds to the case where one has measurement errors that needs to be filtered. Finally, some simulations are presented for a selected few Stiefel manifolds and the speed of convergence is studied.

On the Convergence of the Extended Kalman Filter on Stiefel Manifolds when Observing a Constant Particle with Measurement Errors

TL;DR

This work addresses manifold-valued filtering where observations lie on Stiefel manifolds while the latent state evolves in Euclidean space, and proves convergence guarantees for an extended Kalman filter on under a constant-process assumption. It develops a projection-based measurement model , introduces a projected-normal framework, and uses a Padé-based variance approximation to implement a practical EKF on manifolds. The main theoretical contribution shows that when the true process is constant (, ), the EKF mean converges in probability to the true state and the Kalman variance vanishes; proofs rely on projection-mean commutativity and convergence analysis of Kalman gains. Numerical simulations on multiple Stiefel manifolds demonstrate convergence behavior and reveal the impact of measurement noise and dimensionality on convergence speed, supporting the method's practicality for directional and manifold-valued filtering.

Abstract

In this paper we first introduce the setting of filtering on Stiefel manifolds. Then, assuming the underlying system process is constant, the convergence of the extended Kalman filter with Stiefel manifold-valued observations is proved. This corresponds to the case where one has measurement errors that needs to be filtered. Finally, some simulations are presented for a selected few Stiefel manifolds and the speed of convergence is studied.

Paper Structure

This paper contains 8 sections, 5 theorems, 57 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The extended Kalman filter on Stiefel manifolds
  3. Background
  4. The filtering problem
  5. Proofs
  6. Proof of Proposition \ref{['prop:projectionstiefelcommuteswithmean']}
  7. Proof of Theorem \ref{['thm:convofextkalman']}
  8. Simulations

Key Result

Lemma 2.1

Let $X\in \mathop{\mathrm{GL}}\nolimits_{n,k}(\mathbb{R})$ (resp. $\mathop{\mathrm{GL}}\nolimits_{n,k}(\mathbb{C})$) and let $\phi \in O(n)$ (resp. $U(n)$) be arbitrary. Then it holds that

Figures (11)

  • Figure 1: Simulations of the extended Kalman filter on $\mathop{\mathrm{St}}\nolimits_{4,2}(\mathbb{R})$ given in Algorithm \ref{['alg:kalmansphere']}. The intrinsic scalar variance and the normalized experimental distance squared from the filtered mean to the true point over $m$ measurements. In each figure one can see both given $\xi^2=0.1$ and $\xi^2=0.5$.
  • Figure 2: Simulations of the extended Kalman filter on $\mathop{\mathrm{St}}\nolimits_{6,3}(\mathbb{R})$ given in Algorithm \ref{['alg:kalmansphere']}. The intrinsic scalar variance and the normalized experimental distance squared from the filtered mean to the true point over $m$ measurements. In each figure one can see both given $\xi^2=0.1$ and $\xi^2=0.5$.
  • Figure 3: Simulations of the extended Kalman filter on $\mathop{\mathrm{St}}\nolimits_{12,3}(\mathbb{R})$ given in Algorithm \ref{['alg:kalmansphere']}. The intrinsic scalar variance and the normalized experimental distance squared from the filtered mean to the true point over $m$ measurements. In each figure one can see both given $\xi^2=0.1$ and $\xi^2=0.5$.
  • Figure 4: Simulations of the extended Kalman filter on $\mathop{\mathrm{St}}\nolimits_{15,5}(\mathbb{R})$ given in Algorithm \ref{['alg:kalmansphere']}. The intrinsic scalar variance and the normalized experimental distance squared from the filtered mean to the true point over $m$ measurements. In each figure one can see both given $\xi^2=0.1$ and $\xi^2=0.5$.
  • Figure :
  • ...and 6 more figures

Theorems & Definitions (17)

  • Lemma 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • Remark
  • Example 2.5: Maximal scalar variance for $\mathbb{S}^1$
  • Example 2.6: Maximal scalar variance for the even dimensional spheres
  • Example 2.7: Maximal scalar variance for the odd dimensional spheres
  • Conjecture 2.8
  • ...and 7 more