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Learning Linear Dynamics from Bilinear Observations

Yahya Sattar, Yassir Jedra, Sarah Dean

TL;DR

This work considers the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations, and provides a finite time analysis for learning the unknown dynamics matrices.

Abstract

We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations.

Learning Linear Dynamics from Bilinear Observations

TL;DR

This work considers the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations, and provides a finite time analysis for learning the unknown dynamics matrices.

Abstract

We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations.
Paper Structure (25 sections, 10 theorems, 122 equations, 1 figure)

This paper contains 25 sections, 10 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Problem Formulation
  4. Least Squares Estimation
  5. Main Results on Learning Markov Parameters
  6. Data-dependent Error Bounds
  7. Sample Complexity and Persistence of Excitation
  8. Learning State-Space Parameters
  9. Numerical Experiments
  10. Proofs of Main Results
  11. Learning Markov Parameters (Data-Dependent Bounds)
  12. Proof of Theorem \ref{['thrm:learning_markov_par_main']}
  13. Proof of Lemma \ref{['lemma:output_prediction']}
  14. Persistence of Excitation (Data-Independent Bounds)
  15. Two-sided Concentration Bound -- entire spectrum
  16. ...and 10 more sections

Key Result

Theorem 3.1

Fix $\delta \in (0,1)$, and suppose we are given a single trajectory $\{({\boldsymbol{u}}_t,{\boldsymbol{y}}_t)\}_{t=0}^T$ of the system eqn:bilinear sys. Let ${\boldsymbol{\Gamma}}_w^{\infty} := \sum_{i=0}^{\infty} {\boldsymbol{A}}^i {\boldsymbol{\Sigma}}_w ({\boldsymbol{A}}^i)^{\raisebox{\depth}{$

Figures (1)

  • Figure 1: We plot the estimation error $\|{{\boldsymbol{G}} - \hat{{\boldsymbol{G}}}}\|_{F}^2$ over different values of $T, L, \rho({\boldsymbol{A}})$ while fixing $n=5$ and $p=3$. Our plots show a trade-off between the memory of the system captured by $\rho({\boldsymbol{A}})$ and the number of Markov parameters estimated $L$.

Theorems & Definitions (11)

  • Theorem 3.1: Learning Markov Parameters (Data-Dependent)
  • Lemma 3.2: Output Prediction (Data-Dependent)
  • Theorem 3.3: Persistence of Excitation
  • Theorem 3.4: Learning Markov Parameters (Sample Complexity)
  • Theorem 4.1: Balanced Realization oymak2021revisiting
  • Lemma 6.1: Auto-covariance of effective noise
  • Theorem 6.2: Conditional Covariance of $\Delta{\boldsymbol{G}}$
  • Lemma 6.3: Conditional Mean of $\Delta{\boldsymbol{G}}$
  • Proposition 6.4: Persistence of Excitation
  • Remark 6.3
  • ...and 1 more