High-Dimensional Dynamic Systems Identification with Additional Constraints

TL;DR

This paper develops a unified framework for identifying high-dimensional dynamical systems with low-rank constraints, using nuclear norm heuristics under both noiseless and noisy conditions, with theoretical guarantees.

Contribution

It introduces a weak restricted isometry property and operator norm curvature condition, extending low-rank system identification results to high-dimensional settings with fewer samples.

Findings

Exact low-rank recovery in noiseless case with high probability

Operator norm error bounds in noisy case with finite samples

Extension of existing results to high-dimensional dynamic systems

Abstract

This note presents a unified analysis of the identification of dynamical systems with low-rank constraints under high-dimensional scaling. This identification problem for dynamic systems are challenging due to the intrinsic dependency of the data. To alleviate this problem, we first formulate this identification problem into a multivariate linear regression problem with row-sub-Gaussian measurement matrix using the more general input designs and the independent repeated sampling schemes. We then propose a nuclear norm heuristic method that estimates the parameter matrix of dynamic system from a few input-state data samples. Based on this, we can extend the existing results. In this paper, we consider two scenarios. (i) In the noiseless scenario, nuclear-norm minimization is introduced for promoting low-rank. We define the notion of weak restricted isometry property, which is weaker than the ordinary restricted isometry property, and show it holds with high probability for the row-sub-Gaussian measurement matrix. Thereby, the rank-minimization matrix can be exactly recovered from finite number of data samples. (ii) In the noisy scenario, a regularized framework involving nuclear norm penalty is established. We give the notion of operator norm curvature condition for the loss function, and show it holds for row-sub-Gaussian measurement matrix with high probability. Consequently, when specifying the suitable choice of the regularization parameter, the operator norm error of the optimal solution of this program has a sharp bound given a finite amount of data samples. This operator norm error bound is stronger than the ordinary Frobenius norm error bound obtained in the existing work.