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Online Subspace Learning on Flag Manifolds for System Identification

Dian Jin, Jeremy Coulson

TL;DR

This work proposes a novel online subspace learning framework that operates on flag manifolds and leverages streaming data to recursively track an ensemble of nested subspaces, allowing it to adapt to varying system dimensions without prior knowledge of the true model order.

Abstract

Data-driven control methods based on subspace representations are powerful but are often limited to linear time-invariant systems where the model order is known. A key challenge is developing online data-driven control algorithms for time-varying systems, especially when the system's complexity is unknown or changes over time. To address this, we propose a novel online subspace learning framework that operates on flag manifolds. Our algorithm leverages streaming data to recursively track an ensemble of nested subspaces, allowing it to adapt to varying system dimensions without prior knowledge of the true model order. We show that our algorithm is a generalization of the Grassmannian Recursive Algorithm for Tracking. The learned subspace models are then integrated into a data-driven simulation framework to perform prediction for unknown dynamical systems. The effectiveness of this approach is demonstrated through a case study where the proposed adaptive predictor successfully handles abrupt changes in system dynamics and outperforms several baselines.

Online Subspace Learning on Flag Manifolds for System Identification

TL;DR

This work proposes a novel online subspace learning framework that operates on flag manifolds and leverages streaming data to recursively track an ensemble of nested subspaces, allowing it to adapt to varying system dimensions without prior knowledge of the true model order.

Abstract

Data-driven control methods based on subspace representations are powerful but are often limited to linear time-invariant systems where the model order is known. A key challenge is developing online data-driven control algorithms for time-varying systems, especially when the system's complexity is unknown or changes over time. To address this, we propose a novel online subspace learning framework that operates on flag manifolds. Our algorithm leverages streaming data to recursively track an ensemble of nested subspaces, allowing it to adapt to varying system dimensions without prior knowledge of the true model order. We show that our algorithm is a generalization of the Grassmannian Recursive Algorithm for Tracking. The learned subspace models are then integrated into a data-driven simulation framework to perform prediction for unknown dynamical systems. The effectiveness of this approach is demonstrated through a case study where the proposed adaptive predictor successfully handles abrupt changes in system dynamics and outperforms several baselines.

Paper Structure

This paper contains 17 sections, 1 theorem, 10 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Preliminaries
  4. Subspace geometry
  5. Problem Setup & Motivation
  6. Motivating Example: Online System Identification
  7. Online subspace learning on flag manifolds
  8. Gradient descent on flag manifolds with streaming data
  9. Learning time- and dimension-varying subspaces
  10. Numerical case studies
  11. Geodesic tracking
  12. Online data-driven prediction for a switched system
  13. Offline Data Collection.
  14. Subspace Predictor.
  15. Ensemble of Nested Subspace Prediction
  16. ...and 2 more sections

Key Result

proposition 1

Let $\mathcal{A}_1: \mathrm{Flag}(p, (q)) \to \mathrm{Flag}(p, (q))$ be defined by (eq:flag_update), and $\mathcal{A}_2:\mathrm{Gr}(p,q) \to \mathrm{Gr}(p,q)$ be defined by the update rule in Algorithm alg:GREAT. Then, $\mathcal{A}_1$ is equivalent to $\mathcal{A}_2$.

Figures (5)

  • Figure 1: Average chordal distance for different window sizes across 100 experiments.
  • Figure 2: Median cumulative prediction errors with interquartile whiskers: a single flag evaluated at different ranks (orange) versus individually learned Grassmann subspaces (teal).
  • Figure 3: True and predicted trajectories. Left: no learning baseline. Right: the ensemble-$(9, 10)$ prediction model adapts and tracks the system after $t=T_\text{switch}$ quickly.
  • Figure 4: Median cumulative prediction error ($y$-axis) versus varying NSR ($x$-axis).
  • Figure : FRONT

Theorems & Definitions (4)

  • definition 1
  • definition 2
  • proposition 1
  • proof