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Autoregressive Processes on Stiefel and Grassmann Manifolds

Jordi-Lluís Figueras, Aron Persson

Abstract

System identification of autoregressive processes on Stiefel and Grassmann manifolds are presented and studied. We define the system parameters as elements in the orthogonal group and we show that the system can be estimated by averaging over observations. Then we propose an algorithm on how to compute these system parameters using conjugate gradient descent on Stiefel and Grassmann manifolds, respectively.

Autoregressive Processes on Stiefel and Grassmann Manifolds

Abstract

System identification of autoregressive processes on Stiefel and Grassmann manifolds are presented and studied. We define the system parameters as elements in the orthogonal group and we show that the system can be estimated by averaging over observations. Then we propose an algorithm on how to compute these system parameters using conjugate gradient descent on Stiefel and Grassmann manifolds, respectively.

Paper Structure

This paper contains 9 sections, 5 theorems, 75 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Statistics on Riemannian Manifolds
  3. AR-Processes on Homogeneous Spaces by Lie Group Action
  4. Conjugate Gradient descent to solve system identification
  5. Orthogonal/Unitary groups
  6. Stiefel Manifolds
  7. Grassmann Manifolds
  8. Basic Riemannian Geometry
  9. Simulations

Key Result

Proposition 3.1

Suppose $Z_\ell$ is an AR($N$)-process on $M$ as in Definition def:arprocess with system parameters $A_j$. Then, for every $1\leq j\leq N$ and any pair of fixed points $Z_{\ell-j},Z_{\ell-j-1}\in M$ there is a $\Phi_j\in G$ such that

Figures (9)

  • Figure 1: Simulation an AR(1) process using barycenters on $O_{20}$ where the noise is of approximate amplitude $10^{-1}$.
  • Figure 2: Simulation of the gradient descent method on $\mathop{\mathrm{St}}\nolimits_{15,5}$ where the noise is of approximate amplitude $10^{-3}$.
  • Figure 3: Simulation of the gradient descent method on $\mathop{\mathrm{St}}\nolimits_{20,5}$ over different sizes of the noise.
  • Figure 4: Simulation of the gradient descent method on $\mathop{\mathrm{St}}\nolimits_{d,5}$ over different $d$ where the noise is of approximate amplitude $10^{-3}$.
  • Figure 5: Simulation of the gradient descent method on $\mathop{\mathrm{St}}\nolimits_{20,k}$ over different $k$ where the noise is of approximate amplitude $10^{-3}$.
  • ...and 4 more figures

Theorems & Definitions (16)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Remark
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 6 more