# Outlier-Robust Linear System Identification Under Heavy-tailed Noise

arXiv: https://arxiv.org/abs/2501.00421 · HTML: https://mlsanity.com/paper/2501.00421/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 31 Dec 2024.

## Builds on (most cited first, 15 of 15)

- [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)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (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)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)
- [Learning Sparse Dynamical Systems from a Single Sample Trajectory](https://mlsanity.com/paper/1904.09396/) (2019)
- [A Tutorial on Concentration Bounds for System Identification](https://mlsanity.com/paper/1906.11395/) (2019)
- [Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectories](https://mlsanity.com/paper/2203.12794/) (2022)
- [Linear System Identification Under Multiplicative Noise from Multiple Trajectory Data](https://mlsanity.com/paper/2002.06613/) (2020)

## Built on by (most cited first, 4 of 4)

- [Boosting-Enabled Robust System Identification of Partially Observed LTI Systems Under Heavy-Tailed Noise](https://mlsanity.com/paper/2504.18444/) (2025)
- [Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis](https://mlsanity.com/paper/2508.17142/) (2025)
- [On Uniform Error Bounds for Kernel Regression under Non-Gaussian Noise](https://mlsanity.com/paper/2605.09757/) (2026)
- [Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory](https://mlsanity.com/paper/2610.00637/) (2026)

## Abstract (arXiv)

We consider the problem of estimating the state transition matrix of a linear time-invariant (LTI) system, given access to multiple independent trajectories sampled from the system. Several recent papers have conducted a non-asymptotic analysis of this problem, relying crucially on the assumption that the process noise is either Gaussian or sub-Gaussian, i.e., "light-tailed". In sharp contrast, we work under a significantly weaker noise model, assuming nothing more than the existence of the fourth moment of the noise distribution. For this setting, we provide the first set of results demonstrating that one can obtain sample-complexity bounds for linear system identification that are nearly of the same order as under sub-Gaussian noise. To achieve such results, we develop a novel robust system identification algorithm that relies on constructing multiple weakly-concentrated estimators, and then boosting their performance using suitable tools from high-dimensional robust statistics. Interestingly, our analysis reveals how the kurtosis of the noise distribution, a measure of heavy-tailedness, affects the number of trajectories needed to achieve desired estimation error bounds. Finally, we show that our algorithm and analysis technique can be easily extended to account for scenarios where an adversary can arbitrarily corrupt a small fraction of the collected trajectory data. Our work takes the first steps towards building a robust statistical learning theory for control under non-ideal assumptions on the data-generating process.