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Nonexistence of finite-dimensional estimation algebras on closed smooth manifolds

Jiayi Kang, Andrew Salmon, Stephen Shing-Toung Yau

Abstract

Estimation algebras have been extensively studied in Euclidean space, where finite-dimensional estimation algebras form the foundation of the Kalman and Benes filters, and have contributed to the discovery of many other finite-dimensional filters. This work extends the theory of estimation algebras to filtering problems on Riemannian manifolds in continuous time. Our main result demonstrates that, with non-constant observation functions, the estimation algebra associated with the system on closed Riemannian manifolds is infinite-dimensional.

Nonexistence of finite-dimensional estimation algebras on closed smooth manifolds

Abstract

Estimation algebras have been extensively studied in Euclidean space, where finite-dimensional estimation algebras form the foundation of the Kalman and Benes filters, and have contributed to the discovery of many other finite-dimensional filters. This work extends the theory of estimation algebras to filtering problems on Riemannian manifolds in continuous time. Our main result demonstrates that, with non-constant observation functions, the estimation algebra associated with the system on closed Riemannian manifolds is infinite-dimensional.

Paper Structure

This paper contains 8 sections, 9 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Filtering equations on a Riemannian manifold
  3. Preliminaries and notation
  4. Background on filtering on closed manifolds
  5. The Riemannian metric attached to a non-degenerate diffusion process
  6. The DMZ equation on Riemannian manifolds
  7. Nonexistence of finite-dimensional estimation algebras
  8. Conclusion

Key Result

Theorem 1

Let $M$ be a closed smooth manifold, and let $L$ be a diffusion process on $M$ with a second-order term that is non-degenerate everywhere (in the sense of Definition def: non-degenerate diffusion process). Let $h$ be the observation function of a signal-observation model as in Equation eq main theor

Theorems & Definitions (20)

  • Theorem 1: Main Theorem
  • Theorem 2: Main Theorem II
  • Definition 1
  • Lemma 2
  • proof
  • Remark 3: Relation to continuous filtering literature in Euclidean space
  • Theorem 4: Theorem 3.24 in bain2009fundamentals, or in gyongy1997stochastic
  • Definition 5: $H^k(M)$ on Riemannian Manifolds
  • Theorem 6: Theorem 3.1 in gyongy1997stochastic
  • Proposition 7
  • ...and 10 more