  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.
