  One way to understand time-series data is to identify the underlying
dynamical system which generates it. This task can be done by selecting an
appropriate model and a set of parameters which best fits the dynamics while
providing the simplest representation (i.e. the smallest amount of terms). One
such approach is the sparse identification of nonlinear dynamics framework [6]
which uses a sparsity-promoting algorithm that iterates between a partial
least-squares fit and a thresholding (sparsity-promoting) step. In this work,
we provide some theoretical results on the behavior and convergence of the
algorithm proposed in [6]. In particular, we prove that the algorithm
approximates local minimizers of an unconstrained $\ell^0$-penalized
least-squares problem. From this, we provide sufficient conditions for general
convergence, rate of convergence, and conditions for one-step recovery.
Examples illustrate that the rates of convergence are sharp. In addition, our
results extend to other algorithms related to the algorithm in [6], and provide
theoretical verification to several observed phenomena.
