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Minimum distance classification for nonlinear dynamical systems

Dominique Martinez

TL;DR

Dynafit introduces a kernel-based distance metric between observed trajectories and underlying nonlinear dynamics by learning the dual of the Koopman operator, allowing global linearization in a high-dimensional feature space without explicit mappings. The method relies on the eigen-decomposition of the kernel Gram matrix to estimate the operator projection and compute distances for training and test data, enabling efficient one- or multi-class classification of dynamical systems. Kernel engineering permits incorporating prior dynamical knowledge, with a logistic-map kernel illustrating efficient, scalable computation. Empirically, Dynafit achieves competitive accuracy on chaos detection, handwriting dynamics, and dynamic textures while offering substantial training speed advantages over SVM and comparable performance to deep nets.

Abstract

We address the problem of classifying trajectory data generated by some nonlinear dynamics, where each class corresponds to a distinct dynamical system. We propose Dynafit, a kernel-based method for learning a distance metric between training trajectories and the underlying dynamics. New observations are assigned to the class with the most similar dynamics according to the learned metric. The learning algorithm approximates the Koopman operator which globally linearizes the dynamics in a (potentially infinite) feature space associated with a kernel function. The distance metric is computed in feature space independently of its dimensionality by using the kernel trick common in machine learning. We also show that the kernel function can be tailored to incorporate partial knowledge of the dynamics when available. Dynafit is applicable to various classification tasks involving nonlinear dynamical systems and sensors. We illustrate its effectiveness on three examples: chaos detection with the logistic map, recognition of handwritten dynamics and of visual dynamic textures.

Minimum distance classification for nonlinear dynamical systems

TL;DR

Dynafit introduces a kernel-based distance metric between observed trajectories and underlying nonlinear dynamics by learning the dual of the Koopman operator, allowing global linearization in a high-dimensional feature space without explicit mappings. The method relies on the eigen-decomposition of the kernel Gram matrix to estimate the operator projection and compute distances for training and test data, enabling efficient one- or multi-class classification of dynamical systems. Kernel engineering permits incorporating prior dynamical knowledge, with a logistic-map kernel illustrating efficient, scalable computation. Empirically, Dynafit achieves competitive accuracy on chaos detection, handwriting dynamics, and dynamic textures while offering substantial training speed advantages over SVM and comparable performance to deep nets.

Abstract

We address the problem of classifying trajectory data generated by some nonlinear dynamics, where each class corresponds to a distinct dynamical system. We propose Dynafit, a kernel-based method for learning a distance metric between training trajectories and the underlying dynamics. New observations are assigned to the class with the most similar dynamics according to the learned metric. The learning algorithm approximates the Koopman operator which globally linearizes the dynamics in a (potentially infinite) feature space associated with a kernel function. The distance metric is computed in feature space independently of its dimensionality by using the kernel trick common in machine learning. We also show that the kernel function can be tailored to incorporate partial knowledge of the dynamics when available. Dynafit is applicable to various classification tasks involving nonlinear dynamical systems and sensors. We illustrate its effectiveness on three examples: chaos detection with the logistic map, recognition of handwritten dynamics and of visual dynamic textures.
Paper Structure (7 sections, 19 equations, 3 figures, 2 tables)

This paper contains 7 sections, 19 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Dynafit learning
  4. Dynafit inference
  5. Kernel engineering
  6. Results
  7. Conclusion

Figures (3)

  • Figure 1: Illustration of distance-based classification. A). In traditional classifiers, the distance is a measure of (dis)similarity between a test sample (here the image of Picasso's signature) and a stored prototype (Pissarro's signature). B). In Dynafit, the distance is between a trajectory sample (here the handwritten dynamics) and the dynamical system that has potentially generated the data.
  • Figure 2: Reproduction of characters with the first $N=$10 (plots in red), 20 (plots in green), 50 (plots in blue) and 100 (plots in black) sensor data sampled during the pen trajectory. The data are the pen velocity in x-position (mm/s), pen velocity in y-position (mm/s) and rate of the pen tip force (N/s) recorded at 200 Hz.
  • Figure 3: Image samples from the UCLA dynamic texture dataset (50 classes of different dynamic textures). Each pannel corresponds to the first image in the test sequence of a given class.