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Extended Kalman Filtering on Stiefel Manifolds

Jordi-Lluís Figueras, Aron Persson, Lauri Viitasaari

TL;DR

The paper addresses state estimation for systems whose state evolves in $\mathbb{R}^{n\times k}$ but is observed only through projections onto the Stiefel manifold $\mathrm{St}_{n,k}$. It develops an Extended Kalman Filter on Stiefel manifolds by leveraging the tangent-space geometry, the Riemannian exponential/log maps, and a function $\eta$ that relates ambient Gaussian variance to intrinsic scalar variance, enabling manifold-aware prediction and update steps. The authors present a concrete EKF formulation and provide simulations on $\mathbb{S}^2$ and $\mathrm{St}_{4,2}$ showing substantial improvements over raw measurements, illustrating practical gains in manifold-valued state estimation. The work advances directionally statistical filtering by integrating intrinsic manifold notions into Kalman-type updates, with implications for robotics, communications, and directional data analysis. $

Abstract

A generalisation of the extended Kalman filter for Stiefel manifold-valued measurements is presented. We provide simulations on the 2-sphere and the space of orthogonal 4-by-2 matrices which show significant improvement of the Extended Kalman Filter compared to only relying on raw measurements.

Extended Kalman Filtering on Stiefel Manifolds

TL;DR

The paper addresses state estimation for systems whose state evolves in but is observed only through projections onto the Stiefel manifold . It develops an Extended Kalman Filter on Stiefel manifolds by leveraging the tangent-space geometry, the Riemannian exponential/log maps, and a function that relates ambient Gaussian variance to intrinsic scalar variance, enabling manifold-aware prediction and update steps. The authors present a concrete EKF formulation and provide simulations on and showing substantial improvements over raw measurements, illustrating practical gains in manifold-valued state estimation. The work advances directionally statistical filtering by integrating intrinsic manifold notions into Kalman-type updates, with implications for robotics, communications, and directional data analysis. $

Abstract

A generalisation of the extended Kalman filter for Stiefel manifold-valued measurements is presented. We provide simulations on the 2-sphere and the space of orthogonal 4-by-2 matrices which show significant improvement of the Extended Kalman Filter compared to only relying on raw measurements.

Paper Structure

This paper contains 8 sections, 24 equations, 21 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Examples of projected dynamics
  3. Organisation of the paper
  4. Setting
  5. Extended Kalman Filtering on Stiefel Manifolds
  6. Simulations on $\mathbb{S}^2$
  7. Simulations on $\mathop{\mathrm{St}}\nolimits_{4,2}$
  8. Discussion

Figures (21)

  • Figure 1: A picture of a stochastic process which is then non-linearly projected onto a non-linear space.
  • Figure 2: A picture of a stochastic process $X_t$ that is (non-linearly) projected onto a manifold together with some measurements $z_1,z_2,z_3$
  • Figure 3: A conceptual picture of the tangent space at a point $X\in \mathop{\mathrm{St}}\nolimits_{n,k}$, together with two tangent vectors $V,W\in T_{X} \mathop{\mathrm{St}}\nolimits_{n,k}$ and their corresponding curves that has the corresponding momentous velocity at $X$.
  • Figure 4: An illustration of the exponential map and how it maps tangent vectors $V,W\in T_X\mathop{\mathrm{St}}\nolimits_{n,k}$ to points on $\mathop{\mathrm{St}}\nolimits_{n,k}$.
  • Figure 5: A picture on how the inherent average $\bar{Y}$ relates to a set of points $Y_1,Y_2,Y_3,Y_4\in \mathop{\mathrm{St}}\nolimits_{n,k}$.
  • ...and 16 more figures

Theorems & Definitions (4)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Conjecture 2.4