# Joint Learning of Linear Time-Invariant Dynamical Systems

arXiv: https://arxiv.org/abs/2112.10955 · HTML: https://mlsanity.com/paper/2112.10955/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 21 Dec 2021.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Near-optimal Representation Learning for Linear Bandits and Linear RL](https://mlsanity.com/paper/2102.04132/) (2021)
- [On the Power of Multitask Representation Learning in Linear MDP](https://mlsanity.com/paper/2106.08053/) (2021)
- [A Unified Computational and Statistical Framework for Nonconvex Low-Rank Matrix Estimation](https://mlsanity.com/paper/1610.05275/) (2016)
- [On Adaptive Linear-Quadratic Regulators](https://mlsanity.com/paper/1806.10749/) (2018)
- [Regret Bounds for Lifelong Learning](https://mlsanity.com/paper/1610.08628/) (2016)
- [Input Perturbations for Adaptive Control and Learning](https://mlsanity.com/paper/1811.04258/) (2018)
- [Convergence Properties of Adaptive Systems and the Definition of Exponential Stability](https://mlsanity.com/paper/1511.03222/) (2015)
- [Factor Models with Real Data: a Robust Estimation of the Number of Factors](https://mlsanity.com/paper/1709.01168/) (2017)
- [Scalable regret for learning to control network-coupled subsystems with unknown dynamics](https://mlsanity.com/paper/2108.07970/) (2021)

## Built on by (most cited first, 16 of 16)

- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [Multi-Task Imitation Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2212.00186/) (2022)
- [Nearly Minimax Algorithms for Linear Bandits with Shared Representation](https://mlsanity.com/paper/2203.15664/) (2022)
- [Learning Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2201.11211/) (2022)
- [Sample-Efficient Linear Representation Learning from Non-IID Non-Isotropic Data](https://mlsanity.com/paper/2308.04428/) (2023)
- [Learning Dynamical Systems by Leveraging Data from Similar Systems](https://mlsanity.com/paper/2302.04344/) (2023)
- [Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification](https://mlsanity.com/paper/2401.00073/) (2023)
- [Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control](https://mlsanity.com/paper/2407.05781/) (2024)
- [Joint Learning-Based Stabilization of Multiple Unknown Linear Systems](https://mlsanity.com/paper/2201.01387/) (2022)
- [Combining Federated Learning and Control: A Survey](https://mlsanity.com/paper/2407.11069/) (2024)
- [Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples](https://mlsanity.com/paper/2410.11227/) (2024)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)
- [Finite Sample Analysis of Tensor Decomposition for Learning Mixtures of Linear Systems](https://mlsanity.com/paper/2412.10615/) (2024)
- [Harnessing Data from Clustered LQR Systems: Personalized and Collaborative Policy Optimization](https://mlsanity.com/paper/2511.17489/) (2025)
- [Joint Learning of Linear Dynamical Systems under Smoothness Constraints](https://mlsanity.com/paper/2406.01094/) (2024)
- [On the Effect of Instability on Learning Continuous-Time Linear Control Systems](https://mlsanity.com/paper/2409.11327/) (2024)

## Abstract (arXiv)

Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related linear systems to estimate their transition matrices more accurately. To address this problem, the current paper investigates methods for jointly estimating the transition matrices of multiple systems. It is assumed that the transition matrices are unknown linear functions of some unknown shared basis matrices. We establish finite-time estimation error rates that fully reflect the roles of trajectory lengths, dimension, and number of systems under consideration. The presented results are fairly general and show the significant gains that can be achieved by pooling data across systems in comparison to learning each system individually. Further, they are shown to be robust against model misspecifications. To obtain the results, we develop novel techniques that are of interest for addressing similar joint-learning problems. They include tightly bounding estimation errors in terms of the eigen-structures of transition matrices, establishing sharp high probability bounds for singular values of dependent random matrices, and capturing effects of misspecified transition matrices as the systems evolve over time.