# Finite Sample Analysis of Tensor Decomposition for Learning Mixtures of Linear Systems

arXiv: https://arxiv.org/abs/2412.10615 · HTML: https://mlsanity.com/paper/2412.10615/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 13 Dec 2024.

## Builds on (most cited first, 9 of 9)

- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Learning Mixtures of Linear Regressions with Nearly Optimal Complexity](https://mlsanity.com/paper/1802.07895/) (2018)
- [Solving a Mixture of Many Random Linear Equations by Tensor Decomposition and Alternating Minimization](https://mlsanity.com/paper/1608.05749/) (2016)
- [Multi-Task Imitation Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2212.00186/) (2022)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Joint Learning of Linear Time-Invariant Dynamical Systems](https://mlsanity.com/paper/2112.10955/) (2021)
- [Learning Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2201.11211/) (2022)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [Score Function Features for Discriminative Learning: Matrix and Tensor Framework](https://mlsanity.com/paper/1412.2863/) (2014)

## Built on by (most cited first, 1 of 1)

- [Learning Mixtures of Linear Dynamical Systems via Hybrid Tensor-EM Method](https://mlsanity.com/paper/2510.06091/) (2025)

## Abstract (arXiv)

We study the problem of learning mixtures of linear dynamical systems (MLDS) from input-output data. The mixture setting allows us to leverage observations from related dynamical systems to improve the estimation of individual models. Building on spectral methods for mixtures of linear regressions, we propose a moment-based estimator that uses tensor decomposition to estimate the impulse response parameters of the mixture models. The estimator improves upon existing tensor decomposition approaches for MLDS by utilizing the entire length of the observed trajectories. We provide sample complexity bounds for estimating MLDS in the presence of noise, in terms of both the number of trajectories N and the trajectory length T, and demonstrate the performance of the estimator through simulations.