M.Sc Student M.Sc Thesis Rahamim Ohad Aligning Sets of Temporal Signals with Riemannian Geometry and Koopman Operator Department of Electrical and Computer Engineering ASSOCIATE PROF. Ronen Talmon Abstract

Aligning sets of measurements is a longstanding problem in applied science. Here, we focus on aligning data sets of short temporal signals. Broadly, when analyzing signals in the time domain, it is often useful to view the signals as observations of a dynamical system. Here, we take advantage of a recent line of methods based on Koopman operator theory that attempt to learn the dynamical system from observations in a model-free manner. The Koopman operator is an infinite-dimensional linear operator that represents a possibly non-linear dynamical system by describing the linear evolution of functions defined on the state space.

Over the years, several methods to construct a finite-dimensional operator (matrix) that approximates the Koopman operator from system observations have been proposed. Perhaps the most commonly-used is the dynamic mode decomposition (DMD), further extensions of the DMD algorithm manage to compute a more numerically stable approximation using the singular value decomposition (SVD) and a more accurate approximation using delay-coordinates.

In this research, we present a two-step method for aligning data sets of short temporal signals. In the first step, we compute a DMD matrix approximation of the Koopman operator of each short temporal signal in the data sets at hand. These matrices are viewed as features that capture the dynamical information of the signals. In the second stage, we define a symmetric positive-definite (SPD) matrix based on each DMD matrix. The Riemannian geometry of SPD matrices has been studied for many years. Presumably, the fact that many basic operations, such as the logarithmic and exponential maps, and the Riemannian distance have explicit closed-form expressions has made SPD matrices convenient and useful features. Indeed, SPD matrices have been commonly used, and their Riemannian geometry has been incorporated in computational methods for various tasks, often achieving state of the art performance. Here, we propose to use parallel transport (PT) of SPD matrices, which also has an explicit closed-form expression and can be efficiently implemented in order to align sets of DMD matrices. Specifically, in order to align two sets, we propose to parallel transport the SPD features of the signals of one set from their Riemannian mean to the Riemannian mean of the other set along the unique geodesic path.

Seemingly, PT is nothing more than the Riemannian counterpart of mean subtraction. Yet, we show that by applying such a PT operation to the SPD features, we attain a useful alignment of the respective DMD matrices in terms of the signal dynamics. In addition, we describe a natural extension for the alignment of more than two sets of signals.

We demonstrate the properties of the proposed method on observations of a simulated mechanical system. In addition, we present two applications: sleep stage identification and pre-epileptic seizure detection, where we show accurate alignment of signals recorded from different subjects in a completely unsupervised, data-driven, and model-free fashion.