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On the Convergence of the SINDy Algorithm
Linan Zhang, Hayden Schaeffer
TL;DR
The paper addresses how to identify governing dynamical equations from time-series data while maintaining sparse representations. It analyzes SINDy’s alternating least-squares and thresholding procedure, proving convergence to local minimizers of an unconstrained ℓ0-penalized least-squares objective. The results establish finite-step convergence and sharp rates, while also identifying scope limits and extending guarantees to related algorithms.
Problem
Dynamic model identification seeks governing equations from time-series data while selecting a sparse representation that avoids overfitting.
Method
The paper analyzes SINDy’s iterative sparse-regression scheme, which alternates least-squares fitting over selected terms with thresholding.
Results
The iterates converge in at most n steps, strictly decrease the objective unless stationary, and converge to a fixed point that is a local minimizer.
Takeaways & Limitations
The results provide theoretical support for SINDy’s observed rapid convergence and extend convergence results to other SINDy-based algorithms.
Takeaways & Limitations
The scheme converges to a local minimizer but does not necessarily obtain all local minima, although the global minimizer is obtainable.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper situates SINDy within sparse dynamical-system identification and develops theoretical guarantees for its alternating least-squares and thresholding procedure.
- Dynamic model identification learns governing differential equations from time-series data across applications including weather, aircraft design, and disease spread.
- SINDy represents the governing equation using candidate functions in a data-driven dictionary and estimates sparse coefficients by fitting time derivatives.The dictionary is assumed sufficiently large to capture the nonlinear dynamics.
- The analysis theoretically supports the observed rapid convergence and appropriate sparse approximations produced by SINDy.Examples are used to assess the sharpness of the convergence rates.
- The algorithm alternates between a reduced least-squares problem and a thresholding step to obtain a sparse model and avoid overfitting.Related approaches use ridge regression, LASSO, weak formulations, sampling strategies, and group-sparse hard thresholding.
- This work shows that SINDy approximates local minimizers of an unconstrained ℓ0-penalized least-squares problem and provides convergence conditions and rates.The results also extend to other SINDy-based algorithms.
2 Convergence Analysis
The analysis proves finite-step convergence of the SINDy iterative scheme and characterizes its fixed points as local minimizers of a non-convex objective. It also establishes global-minimizer recoverability, one-step conditions, and extensions to related algorithms.
- Finite-step convergence: The SINDy scheme converges in at most n steps, with nested support sets driving finite termination.If the target is s-sparse, convergence to an s-sparse solution occurs in at most card(S0) −s steps.
- Objective descent: The iterates strictly decrease the objective function unless they are stationary, producing a minimizing sequence for the associated non-convex sparse-approximation objective.The analysis assumes ∥A∥2 = 1 when establishing this objective-based behavior.
- Minimizer characterization: Every fixed point of the scheme is a local minimizer of the objective, while every global minimizer is a fixed point and therefore obtainable by the iteration.The convergence proof establishes approach to a fixed point before relating fixed points to local minimizers.
- One-step recovery: A necessary and sufficient condition is provided for recovering a feasible vector in one step, with the threshold parameter determining immediate convergence.For one example, choosing λ ∈(0.95, 10] yields immediate convergence.
- Extensions: The convergence, minimizer, and descent guarantees extend to the STRidge algorithm and related SINDy-based schemes.For STRidge, fixed points and global minimizers are characterized using its regularized objective F1.
- Examples and sharpness: Examples show that iterating can locate the correct support even when direct thresholding cannot, while the objective value decreases strictly across iterations.One example obtains a 1-sparse approximation in four steps, the maximum required for a 1-sparse solution in that setting.
3 Application: Model Identification of Dynamical Systems
The paper applies SINDy to recover governing equations from noisy time-series data using sparse candidate-function dictionaries. Lorenz and Thomas-system experiments show accurate coefficient recovery and trajectory agreement under moderate noise.
- Application setup: SINDy recovers an unknown governing equation f by finding a sparse coefficient vector in a candidate-function dictionary.The dictionary contains nonlinear candidate functions, while the derivative vector is numerically approximated from observed data.
- Lorenz system: The Lorenz experiment uses polynomial candidate functions up to order p and applies the algorithm with p = 5 and λ = 0.8 at different noise levels.The recovered coefficients and identified systems are reported for two noise levels.
- Lorenz system: σ2 = 0.1 yields trajectory agreement with the Lorenz attractor until about t = 5, whereas σ2 = 0.5 maintains small error only until about t = 4.Increasing noise produces larger parameter and trajectory errors.
- Thomas system: The Thomas-system dictionary combines polynomial, sine, and cosine candidate functions, enabling identification of a non-polynomial chaotic system.The sine and cosine matrices apply these functions elementwise to polynomial features.
- Thomas system: E(x) = 0.0267, and the identified Thomas system is exact up to two significant digits.Its trajectory traces a similar state-space region to the exact trajectory despite coefficient inaccuracies.
4 Discussion
The discussion emphasizes that the analysis characterizes SINDy’s convergence and supports its observed sparse-recovery behavior, while identifying noise effects as an open direction.
- Discussion: The analysis shows that SINDy approximates local minimizers of an ℓ0-penalized least-squares problem and produces a minimizing sequence converging rapidly to a fixed point.The discussion also reports sharp convergence-rate examples and the need for iteration in some recoveries.
- Discussion: Iterating the algorithm can obtain solutions that cannot be obtained by thresholding the least-squares solution once.This establishes a specific role for repeated updates beyond one-step thresholding.
- Discussion: Quantifying the relationship among the thresholding parameter, noise, and expected recovery error remains future work.The paper identifies noise effects as an unresolved aspect of the analysis.
- Discussion: The Thomas-system experiment reports recovered coefficients for two noise levels.The discussion section points to this experiment as part of the paper’s empirical support.