# Exact Recovery Guarantees for Parameterized Nonlinear System Identification Problem under Sparse Disturbances or Semi-Oblivious Attacks

arXiv: https://arxiv.org/abs/2409.00276 · HTML: https://mlsanity.com/paper/2409.00276/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 30 Aug 2024.

## Builds on (most cited first, 17 of 17)

- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [CROP: Certifying Robust Policies for Reinforcement Learning through Functional Smoothing](https://mlsanity.com/paper/2106.09292/) (2021)
- [Policy Smoothing for Provably Robust Reinforcement Learning](https://mlsanity.com/paper/2106.11420/) (2021)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Safely Learning to Control the Constrained Linear Quadratic Regulator](https://mlsanity.com/paper/1809.10121/) (2018)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data](https://mlsanity.com/paper/2001.05484/) (2020)
- [Nonsmooth Analysis and Optimization](https://mlsanity.com/paper/1708.04180/) (2017)
- [Learning of Dynamical Systems under Adversarial Attacks -- Null Space Property Perspective](https://mlsanity.com/paper/2210.01421/) (2022)
- [Analysis of A Nonsmooth Optimization Approach to Robust Estimation](https://mlsanity.com/paper/1402.1899/) (2014)
- [On a class of optimization-based robust estimators](https://mlsanity.com/paper/1705.02837/) (2017)
- [Exact Recovery for System Identification with More Corrupt Data than Clean Data](https://mlsanity.com/paper/2305.10506/) (2023)

## Built on by (most cited first, 1 of 1)

- [Subgradient Method for System Identification with Non-Smooth Objectives](https://mlsanity.com/paper/2503.16673/) (2025)

## Abstract (arXiv)

In this work, we study the problem of learning a nonlinear dynamical system by parameterizing its dynamics using basis functions. We assume that disturbances occur at each time step with an arbitrary probability p, which models the sparsity level of the disturbance vectors over time. These disturbances are drawn from an arbitrary, unknown probability distribution, which may depend on past disturbances, provided that it satisfies a zero-mean assumption. The primary objective of this paper is to learn the system's dynamics within a finite time and analyze the sample complexity as a function of p. To achieve this, we examine a LASSO-type non-smooth estimator, and establish necessary and sufficient conditions for its well-specifiedness and the uniqueness of the global solution to the underlying optimization problem. We then provide exact recovery guarantees for the estimator under two distinct conditions: boundedness and Lipschitz continuity of the basis functions. We show that finite-time exact recovery is achieved with high probability, even when p approaches 1. Unlike prior works, which primarily focus on independent and identically distributed (i.i.d.) disturbances and provide only asymptotic guarantees for system learning, this study presents the first finite-time analysis of nonlinear dynamical systems under a highly general disturbance model. Our framework allows for possible temporal correlations in the disturbances and accommodates semi-oblivious adversarial attacks, significantly broadening the scope of existing theoretical results.