Linear Dynamical Systems (LDS) are powerful tools for modeling time-series data across various fields. Their ability to capture the evolution of systems over time makes them invaluable in areas ranging from econometrics to neuroscience. Recently, the challenge of learning mixtures of Linear Dynamical Systems has emerged as a critical area of research, aiming to provide a more nuanced understanding of complex datasets. Pioneering this direction, Ankur Moitra, along with his colleagues, has introduced a novel approach to tackle this problem, leveraging the power of tensor decompositions. This innovative method offers significant advancements in our ability to learn and interpret intricate dynamic systems.
Traditional methods for learning mixtures of LDS often face limitations, particularly when dealing with complex datasets where clear separation between different system components is not present. Ankur Moitra’s work, presented at the 40th International Conference on Machine Learning (ICML), introduces a tensor decomposition-based algorithm that overcomes these challenges. This approach allows for successful learning even without strong separation conditions, marking a significant step forward in the field. The algorithm’s capability extends to competing with Bayes optimal clustering of trajectories, demonstrating its robustness and accuracy in identifying underlying patterns within mixed systems.
A key strength of this research is its applicability to partially-observed settings, a common scenario in real-world data analysis where not all system states are directly measurable. This makes Ankur Moitra’s learning framework particularly relevant for practical applications. The foundation of this innovative algorithm lies in a insightful connection drawn between the classic Ho-Kalman algorithm from control theory and modern tensor decomposition techniques used in latent variable models. By recognizing this relationship, Moitra and his team have developed a “playbook” to extend these methods for learning more complex generative models, opening up new avenues for research and application in learning linear dynamical systems.
In conclusion, Ankur Moitra’s work on learning mixtures of linear dynamical systems using tensor decompositions represents a significant contribution to the field. By offering a robust, accurate, and broadly applicable algorithm, this research enhances our ability to model and understand complex time-series data, paving the way for deeper insights in diverse scientific and engineering domains.
@InProceedings{pmlr-v202-bakshi23a,
title = {Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems},
author = {Bakshi, Ainesh and Liu, Allen and Moitra, Ankur and Yau, Morris},
booktitle = {Proceedings of the 40th International Conference on Machine Learning},
pages = {1549--1563},
year = {2023},
editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan},
volume = {202},
series = {Proceedings of Machine Learning Research},
month = {23--29 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v202/bakshi23a/bakshi23a.pdf},
url = {https://proceedings.mlr.press/v202/bakshi23a.html},
abstract = {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 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.}
}
