  We consider the problem of learning a realization for a linear time-invariant
(LTI) dynamical system from input/output data. Given a single input/output
trajectory, we provide finite time analysis for learning the system's Markov
parameters, from which a balanced realization is obtained using the classical
Ho-Kalman algorithm. By proving a stability result for the Ho-Kalman algorithm
and combining it with the sample complexity results for Markov parameters, we
show how much data is needed to learn a balanced realization of the system up
to a desired accuracy with high probability.
