Efficient Estimation of Strictly Stable Linear Dynamical Systems under Convex Constraints

Core Concepts

The core message of this paper is that incorporating prior structural information about the system matrix A* in the form of a convex constraint set K can lead to significantly improved finite-time identification performance compared to the unconstrained ordinary least squares (OLS) estimator, especially when the intrinsic dimension of A* is much smaller than the ambient dimension n.

Abstract

The paper considers the problem of finite-time identification of a linear dynamical system (LDS) of the form xt+1 = Axt + ηt+1, where A is the unknown system matrix to be estimated from T samples of a single trajectory (xt)Tt=0. The authors assume that A* is strictly stable, i.e., its spectral radius ρ(A*) < 1.
The key contributions are as follows:

The authors derive non-asymptotic error bounds in the Frobenius norm for the constrained least squares estimator of A*, where the constraint set K captures prior structural information about A*. The bounds depend on the local size of K at A*, as measured by Talagrand's γ1 and γ2 functionals.

The authors instantiate their general results for three specific examples of K:

K is a d-dimensional subspace of Rn×n
K is a suitably scaled ℓ1 ball, capturing sparsity of A*
K consists of matrices formed by sampling a bivariate convex function on a uniform grid, capturing convex structure of A*

For these examples, the authors show that the sample complexity T required to accurately estimate A* can be significantly smaller than the unconstrained case, especially when the intrinsic dimension of A* (d, sparsity level k, or complexity of the underlying convex function) is much smaller than the ambient dimension n.

The proofs rely on novel concentration inequalities for suprema of second-order subgaussian chaos processes, which are of independent interest.

Overall, the paper demonstrates the benefits of leveraging structural constraints in the finite-time identification of linear dynamical systems.

Stats

The paper does not contain any explicit numerical data or statistics. The key quantities used in the analysis are:

ρ(A*): Spectral radius of the system matrix A*
J(A*): A quantity defined in (2.1) that captures the stability of A*
γ1, γ2: Talagrand's gamma functionals that measure the local complexity of the constraint set K

Quotes

"We assume prior structural information on A* is available, which can be captured in the form of a convex set K containing A*. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm that depend on the local size of K at A*."
"Upon instantiating our bounds for the aforementioned choices of K, we obtain the following corollaries."

Key Insights Distilled From

Learning linear dynamical systems under convex constraints

by Hemant Tyagi... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2303.15121.pdf

Deeper Inquiries

How can the results be extended to the case where the system matrix A* is only marginally stable (ρ(A*) ≤ 1) or even unstable (ρ(A*) > 1)

To extend the results to the case where the system matrix A* is marginally stable (ρ(A*) ≤ 1) or even unstable (ρ(A*) > 1), we need to adapt the analysis to handle these scenarios.
For the case where A* is marginally stable (ρ(A*) ≤ 1), the analysis can be modified to incorporate the spectral radius constraint. This would involve adjusting the error bounds and sample complexity requirements to account for the different stability condition. The key would be to consider the impact of the spectral radius on the estimation error and tailor the bounds accordingly.
In the case of an unstable system (ρ(A*) > 1), the analysis would need to be significantly altered. Unstable systems pose unique challenges in estimation due to their explosive behavior. Techniques such as robust estimation methods or regularization to stabilize the estimation process could be explored. The error bounds and sample complexity requirements would need to be redefined to address the instability of the system matrix.
Overall, extending the results to cover marginally stable or unstable systems would require a thorough reevaluation of the analysis, taking into account the specific characteristics and challenges posed by these stability conditions.

Can the authors' proof techniques be applied to other structured estimation problems beyond linear dynamical systems, such as sparse or low-rank matrix recovery

The proof techniques employed by the authors for estimating linear dynamical systems under convex constraints can indeed be applied to other structured estimation problems beyond linear dynamical systems, such as sparse or low-rank matrix recovery. The key lies in adapting the analysis to suit the specific structure and constraints of the problem at hand.
For sparse matrix recovery, similar concentration inequalities and optimization techniques can be utilized to derive error bounds and sample complexity requirements. The concept of sparsity can be incorporated into the analysis, and the bounds can be tailored to promote sparsity in the estimated matrix.
In the case of low-rank matrix recovery, the analysis can focus on the low-rank structure of the matrix and utilize techniques that encourage low-rank solutions. Regularization methods that penalize the rank of the estimated matrix can be incorporated, and the error bounds can be derived based on the rank constraint.
Overall, the proof techniques can be generalized to a variety of structured estimation problems by adapting the analysis to suit the specific constraints and characteristics of the problem under consideration.

Are there other types of structural constraints beyond convex sets that could be incorporated in the estimation of linear dynamical systems, and how would the analysis change

While convex sets are a common and powerful constraint for linear dynamical systems estimation, there are other types of structural constraints that could be incorporated into the estimation process. Some examples include:

Low-rank constraints: By imposing a low-rank constraint on the system matrix A*, the estimation process can be guided towards solutions with reduced rank. This constraint is useful in scenarios where the underlying system has a low-rank structure, and it can help in reducing the complexity of the estimation problem.

Group sparsity constraints: Group sparsity constraints encourage certain groups of parameters in the system matrix to be jointly active or inactive. This constraint is beneficial when the parameters exhibit a grouped structure, and it can lead to more interpretable and efficient estimation results.

Graph constraints: Incorporating graph constraints can capture the relationships and dependencies between different elements of the system matrix. Graph-based regularization techniques can promote smoothness or sparsity patterns based on the graph structure, enhancing the estimation process.

The analysis for linear dynamical systems under these alternative structural constraints would need to be tailored to accommodate the specific characteristics of each constraint. The error bounds and sample complexity requirements would be adjusted to reflect the impact of the new constraints on the estimation process.