# Finite Sample Identification of Bilinear Dynamical Systems

arXiv: https://arxiv.org/abs/2208.13915 · HTML: https://mlsanity.com/paper/2208.13915/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 34 references that are in the map.
- First posted on arXiv 29 Aug 2022.

## Builds on (most cited first, 22 of 22)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [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)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [Regret Bounds for Adaptive Nonlinear Control](https://mlsanity.com/paper/2011.13101/) (2020)
- [Learning Sparse Dynamical Systems from a Single Sample Trajectory](https://mlsanity.com/paper/1904.09396/) (2019)
- [Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics](https://mlsanity.com/paper/2010.09961/) (2020)
- [Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems](https://mlsanity.com/paper/2105.11558/) (2021)
- [Stochastic Gradient Descent Learns State Equations with Nonlinear Activations](https://mlsanity.com/paper/1809.03019/) (2018)
- [Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds](https://mlsanity.com/paper/2111.07018/) (2021)
- [Convex Programming for Estimation in Nonlinear Recurrent Models](https://mlsanity.com/paper/1908.09915/) (2019)
- [Efficient learning of hidden state LTI state space models of unknown order](https://mlsanity.com/paper/2202.01625/) (2022)

## Built on by (most cited first, 12 of 12)

- [The noise level in linear regression with dependent data](https://mlsanity.com/paper/2305.11165/) (2023)
- [Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis](https://mlsanity.com/paper/2309.14648/) (2023)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Non-asymptotic System Identification for Linear Systems with Nonlinear Policies](https://mlsanity.com/paper/2306.10369/) (2023)
- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [PAC-Bayes Generalisation Bounds for Dynamical Systems Including Stable RNNs](https://mlsanity.com/paper/2312.09793/) (2023)
- [A Tensor Decomposition Perspective on Second-order RNNs](https://mlsanity.com/paper/2406.05045/) (2024)
- [End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data](https://mlsanity.com/paper/2409.18010/) (2024)
- [High Effort, Low Gain: Fundamental Limits of Active Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2509.11907/) (2025)
- [Learning Nonlinear Couplings in Network of Agents from a Single Sample Trajectory](https://mlsanity.com/paper/2211.11069/) (2022)
- [Non-Asymptotic Bounds for Closed-Loop Identification of Sub-Exponentially Growing Nonlinear Stochastic Systems](https://mlsanity.com/paper/2412.04157/) (2024)
- [Achieving widetildeO(1/ε) Sample Complexity for Bilinear Systems Identification under Bounded Noises](https://mlsanity.com/paper/2603.20819/) (2026)

## Abstract (arXiv)

Bilinear dynamical systems are ubiquitous in many different domains and they can also be used to approximate more general control-affine systems. This motivates the problem of learning bilinear systems from a single trajectory of the system's states and inputs. Under a mild marginal mean-square stability assumption, we identify how much data is needed to estimate the unknown bilinear system up to a desired accuracy with high probability. Our sample complexity and statistical error rates are optimal in terms of the trajectory length, the dimensionality of the system and the input size. Our proof technique relies on an application of martingale small-ball condition. This enables us to correctly capture the properties of the problem, specifically our error rates do not deteriorate with increasing instability. Finally, we show that numerical experiments are well-aligned with our theoretical results.