{ "id": "1805.06445", "version": "v1", "published": "2018-05-16T17:51:14.000Z", "updated": "2018-05-16T17:51:14.000Z", "title": "On the Convergence of the SINDy Algorithm", "authors": [ "Linan Zhang", "Hayden Schaeffer" ], "comment": "24 pages, 4 figures, 3 tables", "categories": [ "math.OC", "cs.IT", "math.IT" ], "abstract": "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.", "revisions": [ { "version": "v1", "updated": "2018-05-16T17:51:14.000Z" } ], "analyses": { "keywords": [ "convergence", "sindy algorithm", "algorithm approximates local minimizers", "partial least-squares fit", "understand time-series data" ], "note": { "typesetting": "TeX", "pages": 24, "language": "en", "license": "arXiv", "status": "editable" } } }