# Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems

arXiv: https://arxiv.org/abs/2307.06538 · HTML: https://mlsanity.com/paper/2307.06538/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 52 references that are in the map.
- First posted on arXiv 13 Jul 2023.

## 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)
- [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)
- [Toeplitz Inverse Covariance-Based Clustering of Multivariate Time Series Data](https://mlsanity.com/paper/1706.03161/) (2017)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Meta-learning for mixed linear regression](https://mlsanity.com/paper/2002.08936/) (2020)
- [Robust Meta-learning for Mixed Linear Regression with Small Batches](https://mlsanity.com/paper/2006.09702/) (2020)
- [Spectral Filtering for General Linear Dynamical Systems](https://mlsanity.com/paper/1802.03981/) (2018)
- [Learning Mixtures of Linear Regressions with Nearly Optimal Complexity](https://mlsanity.com/paper/1802.07895/) (2018)
- [Learning Mixtures of Linear Regressions in Subexponential Time via Fourier Moments](https://mlsanity.com/paper/1912.07629/) (2019)
- [On the Minimax Optimality of the EM Algorithm for Learning Two-Component Mixed Linear Regression](https://mlsanity.com/paper/2006.02601/) (2020)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics](https://mlsanity.com/paper/2206.02972/) (2022)
- [Learning Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2201.11211/) (2022)
- [Nonparametric Finite Time LTI System Identification](https://mlsanity.com/paper/1902.01848/) (2019)
- [A Wasserstein Minimax Framework for Mixed Linear Regression](https://mlsanity.com/paper/2106.07537/) (2021)
- [Algorithmic Foundations for the Diffraction Limit](https://mlsanity.com/paper/2004.07659/) (2020)
- [Super-Resolution Off the Grid](https://mlsanity.com/paper/1509.07943/) (2015)

## Built on by (most cited first, 10 of 10)

- [An Interventional Perspective on Identifiability in Gaussian LTI Systems with Independent Component Analysis](https://mlsanity.com/paper/2311.18048/) (2023)
- [Structure learning of Hamiltonians from real-time evolution](https://mlsanity.com/paper/2405.00082/) (2024)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)
- [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)
- [Finite Sample Analysis of Tensor Decomposition for Learning Mixtures of Linear Systems](https://mlsanity.com/paper/2412.10615/) (2024)
- [Temporal Robustness in Discrete Time Linear Dynamical Systems](https://mlsanity.com/paper/2505.02347/) (2025)
- [Linear Causal Representation Learning by Topological Ordering, Pruning, and Disentanglement](https://mlsanity.com/paper/2509.22553/) (2025)
- [Learning Mixtures of Linear Dynamical Systems via Hybrid Tensor-EM Method](https://mlsanity.com/paper/2510.06091/) (2025)
- [Modeling Dynamic Mixtures of Time-Delay Systems from Streaming Time Series](https://mlsanity.com/paper/2605.26191/) (2026)

## Abstract (arXiv)

Recently Chen and Poor initiated the study of learning mixtures of linear dynamical systems. While linear dynamical systems already have wide-ranging applications in modeling time-series data, using mixture models can lead to a better fit or even a richer understanding of underlying subpopulations represented in the data. In this work we give a new approach to learning mixtures of linear dynamical systems that is based on tensor decompositions. As a result, our algorithm succeeds without strong separation conditions on the components, and can be used to compete with the Bayes optimal clustering of the trajectories. Moreover our algorithm works in the challenging partially-observed setting. Our starting point is the simple but powerful observation that the classic Ho-Kalman algorithm is a close relative of modern tensor decomposition methods for learning latent variable models. This gives us a playbook for how to extend it to work with more complicated generative models.