# Linear Dynamics: Clustering without identification

## TL;DR

This paper introduces a method to identify the eigenspectrum of linear dynamical systems without full system identification, enabling efficient clustering of time series with temporal offsets and varying lengths.

## Contribution

The work presents a novel, computationally efficient algorithm for estimating eigenvalues of linear systems, improving clustering of time series generated from such systems.

## Key findings

- Algorithm accurately estimates eigenvalues without full system identification.
- Improves clustering quality on synthetic and real ECG data.
- Handles temporal offsets and varying lengths in time series.

## Abstract

Linear dynamical systems are a fundamental and powerful parametric model class. However, identifying the parameters of a linear dynamical system is a venerable task, permitting provably efficient solutions only in special cases. This work shows that the eigenspectrum of unknown linear dynamics can be identified without full system identification. We analyze a computationally efficient and provably convergent algorithm to estimate the eigenvalues of the state-transition matrix in a linear dynamical system. When applied to time series clustering, our algorithm can efficiently cluster multi-dimensional time series with temporal offsets and varying lengths, under the assumption that the time series are generated from linear dynamical systems. Evaluating our algorithm on both synthetic data and real electrocardiogram (ECG) signals, we see improvements in clustering quality over existing baselines.