# Joint Problems in Learning Multiple Dynamical Systems

arXiv: https://arxiv.org/abs/2311.02181 · HTML: https://mlsanity.com/paper/2311.02181/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 70 references that are in the map.
- First posted on arXiv 3 Nov 2023.

## Builds on (most cited first, 23 of 23)

- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Ten Steps of EM Suffice for Mixtures of Two Gaussians](https://mlsanity.com/paper/1609.00368/) (2016)
- [Convergence Rates for Gaussian Mixtures of Experts](https://mlsanity.com/paper/1907.04377/) (2019)
- [Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences](https://mlsanity.com/paper/1609.00978/) (2016)
- [Global analysis of Expectation Maximization for mixtures of two Gaussians](https://mlsanity.com/paper/1608.07630/) (2016)
- [Randomly initialized EM algorithm for two-component Gaussian mixture achieves near optimality in O(√n) iterations](https://mlsanity.com/paper/1908.10935/) (2019)
- [Sharp Analysis of Expectation-Maximization for Weakly Identifiable Models](https://mlsanity.com/paper/1902.00194/) (2019)
- [Multi-Task Imitation Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2212.00186/) (2022)
- [Learning Dynamical Systems with Side Information](https://mlsanity.com/paper/2008.10135/) (2020)
- [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)
- [TSSOS: A Moment-SOS hierarchy that exploits term sparsity](https://mlsanity.com/paper/1912.08899/) (2019)
- [Learning Mixtures of Markov Chains and MDPs](https://mlsanity.com/paper/2211.09403/) (2022)
- [Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension](https://mlsanity.com/paper/2003.03210/) (2020)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [Distributed Learning of Finite Gaussian Mixtures](https://mlsanity.com/paper/2010.10412/) (2020)
- [The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures](https://mlsanity.com/paper/2103.15653/) (2021)
- [Global Optimization of Mixed-Integer Nonlinear Programs with SCIP 8](https://mlsanity.com/paper/2301.00587/) (2023)
- [Sparse Noncommutative Polynomial Optimization](https://mlsanity.com/paper/1909.00569/) (2019)
- [Branch-and-Bound Solves Random Binary IPs in Polytime](https://mlsanity.com/paper/2007.15192/) (2020)
- [SparseJSR: A Fast Algorithm to Compute Joint Spectral Radius via Sparse SOS Decompositions](https://mlsanity.com/paper/2008.11441/) (2020)
- [Linear Dynamics: Clustering without identification](https://mlsanity.com/paper/1908.01039/) (2019)
- [Learning of Linear Dynamical Systems as a Non-Commutative Polynomial Optimization Problem](https://mlsanity.com/paper/2002.01444/) (2020)

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

- [Causal Learning in Biomedical Applications: Krebs Cycle as a Benchmark](https://mlsanity.com/paper/2406.15189/) (2024)

## Abstract (arXiv)

Clustering of time series is a well-studied problem, with applications ranging from quantitative, personalized models of metabolism obtained from metabolite concentrations to state discrimination in quantum information theory. We consider a variant, where given a set of trajectories and a number of parts, we jointly partition the set of trajectories and learn linear dynamical system (LDS) models for each part, so as to minimize the maximum error across all the models. We present globally convergent methods and EM heuristics, accompanied by promising computational results. The key highlight of this method is that it does not require a predefined hidden state dimension but instead provides an upper bound. Additionally, it offers guidance for determining regularization in the system identification.