Improving Prediction Accuracy of Noisy Hankel Matrix Models by Increasing Depth

Core Concepts

Increasing the depth (number of rows) of Hankel matrices constructed from noisy input-output data can significantly improve the accuracy of self-consistent predictions, mitigating the effect of measurement noise.

Abstract

The paper investigates the approximation properties of noisy Hankel matrix models, which arise naturally in the context of linear time-invariant (LTI) systems. When the input and output data are subject to measurement noise, the authors analyze the interplay between the amount of data, the depth of the Hankel matrices, and the overall error in the model.
The key findings are:

The singular values of random Hankel matrices diverge to infinity as the amount of data increases, but this rate can be accelerated by increasing the depth of the matrices.

Applying this insight to the full noisy model, the authors show that increasing the depth mitigates the effect of the noise term, leading to improved self-consistency in the model's predictions.

Numerical experiments demonstrate that simply reconfiguring the Hankel matrices to use a parsimonious depth has a dramatic positive impact on rollout performance, even without any data preprocessing. This depth selection is also shown to be useful for data-driven LQR control.

The authors conclude that the depth of Hankel matrices is an important parameter to consider when working with noisy input-output data, as it can significantly improve the approximation properties of the resulting model.

Stats

The paper does not provide any specific numerical data or statistics. The key insights are derived through theoretical analysis of the singular values of random Hankel matrices.

Quotes

"lim
N→∞P
1
σmin(HL) ≤ϵN
= 1"
"∥˜
yi −yi+L∥≤∥[b
yL . . . b
yN] α −yi+L∥+ ∥[ωL . . . ωN] α∥"

Key Insights Distilled From

Deep Hankel matrices with random elements

by Nathan P. La... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15512.pdf

Deeper Inquiries

How would the analysis and insights change if the input signal was not Gaussian, but had a different distribution

If the input signal was not Gaussian but had a different distribution, the analysis and insights would need to be adjusted accordingly. The properties of the input signal distribution would impact the behavior of the system and the characteristics of the Hankel matrices. For example, if the input signal had a heavy-tailed distribution, it could lead to outliers in the data, affecting the estimation of the system dynamics. The assumptions made about the input signal being Gaussian, such as its persistency of excitation properties, would no longer hold, potentially requiring different analytical approaches to characterize the system.

What are the implications of these findings for other data-driven control methods beyond LQR, such as model predictive control

The implications of the findings for other data-driven control methods beyond LQR, such as model predictive control (MPC), are significant. The insights gained from analyzing the properties of Hankel matrices and their impact on system identification and control can be applied to various data-driven control techniques. For MPC, understanding the role of data depth and the accuracy of the model in predicting system behavior can lead to improved control performance. By optimizing the depth of the Hankel matrices or other data representations used in data-driven control, practitioners can enhance the robustness and efficiency of their control strategies.

Could the techniques used to analyze the singular values of random Hankel matrices be extended to study the properties of other structured random matrix models in machine learning and control applications

The techniques used to analyze the singular values of random Hankel matrices can be extended to study the properties of other structured random matrix models in machine learning and control applications. Random matrix theory provides a powerful framework for understanding the behavior of complex systems with random components. By applying similar analytical tools to different types of structured random matrices, researchers can gain insights into the approximation properties, robustness, and generalization capabilities of data-driven models. This extension can enhance the understanding of various machine learning algorithms, control strategies, and system identification methods based on random matrix theory principles.