# Variational Inference and Learning of Piecewise-linear Dynamical Systems

arXiv: https://arxiv.org/abs/2006.01668 · HTML: https://mlsanity.com/paper/2006.01668/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 8 of 8)

- [Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2410.14240/) (2024)
- [Continuous-Time Piecewise-Linear Recurrent Neural Networks](https://mlsanity.com/paper/2602.15649/) (2026)
- [Cluster-based Regression using Variational Inference and Applications in Financial Forecasting](https://mlsanity.com/paper/2205.00605/) (2022)
- [Discovering group dynamics in coordinated time series via hierarchical recurrent switching-state models](https://mlsanity.com/paper/2401.14973/) (2024)
- [Adaptive time series forecasting with markovian variance switching](https://mlsanity.com/paper/2402.14684/) (2024)
- [Improvements to Supervised EM Learning of Shared Kernel Models by Feature Space Partitioning](https://mlsanity.com/paper/2205.15304/) (2022)
- [A Competitive Learning Approach for Specialized Models: A Solution for Complex Physical Systems with Distinct Functional Regimes](https://mlsanity.com/paper/2307.10496/) (2023)
- [Portfolio Stress Testing and Value at Risk (VaR) Incorporating Current Market Conditions](https://mlsanity.com/paper/2409.18970/) (2024)

## Abstract (arXiv)

Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise-linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because of their computational complexity increases exponentially with time. In this paper, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms, a filter and a smoother. We show that the model parameters can be split into two sets, static and dynamic parameters, and that the former parameters can be estimated off-line together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to a visual tracking problem, namely head-pose tracking, and we thoroughly compare our algorithm with several state of the art trackers.