# Learning Mixtures of Linear Dynamical Systems

arXiv: https://arxiv.org/abs/2201.11211 · HTML: https://mlsanity.com/paper/2201.11211/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 101 references that are in the map.
- First posted on arXiv 26 Jan 2022.

## Builds on (most cited first, 30 of 54)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Contextual Markov Decision Processes](https://mlsanity.com/paper/1502.02259/) (2015)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [On the Theory of Transfer Learning: The Importance of Task Diversity](https://mlsanity.com/paper/2006.11650/) (2020)
- [Multi-Task Reinforcement Learning with Context-based Representations](https://mlsanity.com/paper/2102.06177/) (2021)
- [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)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Information Theoretic Regret Bounds for Online Nonlinear Control](https://mlsanity.com/paper/2006.12466/) (2020)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Mobile Sensor Data Anonymization](https://mlsanity.com/paper/1810.11546/) (2018)
- [RL for Latent MDPs: Regret Guarantees and a Lower Bound](https://mlsanity.com/paper/2102.04939/) (2021)
- [Derivative-Free Methods for Policy Optimization: Guarantees for Linear Quadratic Systems](https://mlsanity.com/paper/1812.08305/) (2018)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Sample Complexity of Multi-task Reinforcement Learning](https://mlsanity.com/paper/1309.6821/) (2013)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Online Linear Quadratic Control](https://mlsanity.com/paper/1806.07104/) (2018)
- [Continuous Meta-Learning without Tasks](https://mlsanity.com/paper/1912.08866/) (2019)
- [AdaRL: What, Where, and How to Adapt in Transfer Reinforcement Learning](https://mlsanity.com/paper/2107.02729/) (2021)
- [Toeplitz Inverse Covariance-Based Clustering of Multivariate Time Series Data](https://mlsanity.com/paper/1706.03161/) (2017)
- [Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems](https://mlsanity.com/paper/2003.11227/) (2020)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Meta-learning for mixed linear regression](https://mlsanity.com/paper/2002.08936/) (2020)

## Built on by (most cited first, 15 of 15)

- [Learning Mixtures of Markov Chains and MDPs](https://mlsanity.com/paper/2211.09403/) (2022)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [An Interventional Perspective on Identifiability in Gaussian LTI Systems with Independent Component Analysis](https://mlsanity.com/paper/2311.18048/) (2023)
- [Efficient List-Decodable Regression using Batches](https://mlsanity.com/paper/2211.12743/) (2022)
- [Conformal Prediction for Time-series Forecasting with Change Points](https://mlsanity.com/paper/2509.02844/) (2025)
- [Linear Regression using Heterogeneous Data Batches](https://mlsanity.com/paper/2309.01973/) (2023)
- [Interventional Causal Discovery in a Mixture of DAGs](https://mlsanity.com/paper/2406.08666/) (2024)
- [A New Approach to Controlling Linear Dynamical Systems](https://mlsanity.com/paper/2504.03952/) (2025)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)
- [Learning the Topology and Behavior of Discrete Dynamical Systems](https://mlsanity.com/paper/2402.11686/) (2024)
- [Finite Sample Analysis of Tensor Decomposition for Learning Mixtures of Linear Systems](https://mlsanity.com/paper/2412.10615/) (2024)
- [Efficient PAC Learnability of Dynamical Systems Over Multilayer Networks](https://mlsanity.com/paper/2405.06884/) (2024)
- [Learning Mixtures of Linear Dynamical Systems via Hybrid Tensor-EM Method](https://mlsanity.com/paper/2510.06091/) (2025)
- [Uniform error bounds for quantized dynamical models](https://mlsanity.com/paper/2602.15586/) (2026)
- [Modeling Dynamic Mixtures of Time-Delay Systems from Streaming Time Series](https://mlsanity.com/paper/2605.26191/) (2026)

## Abstract (arXiv)

We study the problem of learning a mixture of multiple linear dynamical systems (LDSs) from unlabeled short sample trajectories, each generated by one of the LDS models. Despite the wide applicability of mixture models for time-series data, learning algorithms that come with end-to-end performance guarantees are largely absent from existing literature. There are multiple sources of technical challenges, including but not limited to (1) the presence of latent variables (i.e. the unknown labels of trajectories); (2) the possibility that the sample trajectories might have lengths much smaller than the dimension d of the LDS models; and (3) the complicated temporal dependence inherent to time-series data. To tackle these challenges, we develop a two-stage meta-algorithm, which is guaranteed to efficiently recover each ground-truth LDS model up to error tildeO(√d/T), where T is the total sample size. We validate our theoretical studies with numerical experiments, confirming the efficacy of the proposed algorithm.