High-dimensional dynamic systems identification with additional constraints

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 is 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 a row-sub-Gaussian measurement matrix using the more general input designs and the independently repeated sampling schemes. We then propose a nuclear norm heuristic method that estimates the parameter matrix of a dynamic system from a few input-state data samples. Based on this, we can extend the existing results. In this article, 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 a finite number of data samples. (ii) In the noisy scenario, a regularized framework involving the nuclear norm penalty is established. We give the notion of operator norm curvature condition for the loss function and show it holds for the 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 always smaller than the ordinary Frobenius norm error bound obtained in the existing work.