# Learning Linear Dynamics from Bilinear Observations

arXiv: https://arxiv.org/abs/2409.16499 · HTML: https://mlsanity.com/paper/2409.16499/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 62 references that are in the map.
- First posted on arXiv 24 Sep 2024.

## Builds on (most cited first, 30 of 42)

- [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)
- [Optimal Regularization Can Mitigate Double Descent](https://mlsanity.com/paper/2003.01897/) (2020)
- [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015)
- [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)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [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)
- [Bilinear Bandits with Low-rank Structure](https://mlsanity.com/paper/1901.02470/) (2019)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems](https://mlsanity.com/paper/2003.11227/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Spectral Filtering for General Linear Dynamical Systems](https://mlsanity.com/paper/1802.03981/) (2018)
- [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)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Preference Dynamics Under Personalized Recommendations](https://mlsanity.com/paper/2205.13026/) (2022)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Learning the Linear Quadratic Regulator from Nonlinear Observations](https://mlsanity.com/paper/2010.03799/) (2020)
- [Learning Sparse Dynamical Systems from a Single Sample Trajectory](https://mlsanity.com/paper/1904.09396/) (2019)
- [Robust Guarantees for Perception-Based Control](https://mlsanity.com/paper/1907.03680/) (2019)
- [Rebounding Bandits for Modeling Satiation Effects](https://mlsanity.com/paper/2011.06741/) (2020)
- [Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems](https://mlsanity.com/paper/2105.11558/) (2021)
- [Non-Asymptotic Analysis of Robust Control from Coarse-Grained Identification](https://mlsanity.com/paper/1707.04791/) (2017)
- [Stochastic Gradient Descent Learns State Equations with Nonlinear Activations](https://mlsanity.com/paper/1809.03019/) (2018)

## Built on by (most cited first, 6 of 6)

- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [Sub-optimality of the Separation Principle for Quadratic Control from Bilinear Observations](https://mlsanity.com/paper/2504.11555/) (2025)
- [Explore-then-Commit for Nonstationary Linear Bandits with Latent Dynamics](https://mlsanity.com/paper/2510.16208/) (2025)
- [Dual Control of Linear Systems from Bilinear Observations with Belief Space Model Predictive Control](https://mlsanity.com/paper/2604.24663/) (2026)
- [Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory](https://mlsanity.com/paper/2610.00637/) (2026)
- [Block Optimism for Nonstationary Bandits with Latent Linear Dynamics](https://mlsanity.com/paper/2610.00911/) (2026)

## Abstract (arXiv)

We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations.