טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentYair Or
SubjectGeometric Analysis of Signals and Systems
DepartmentDepartment of Electrical Engineering
Supervisor Professor Ronen Talmon
Full Thesis textFull thesis text - English Version


Abstract

Often in applications, high-dimensional data inherently exhibit a geometric structure, which can be discovered and exploited for a broad range of data analysis tasks. The extraction and analysis of the underlying structure of data have attracted significant research efforts in recent years. We address several aspects of this geometric approach for unsupervised data analysis and processing. Our work consists of contributions that can be divided to three parts.


In the first part we consider the problem of unsupervised geometric domain adaptation. Classical domain adaptation problems arise when data acquisition is performed at multiple sites, with different equipment, and under various settings, etc. As a result, the collected datasets suffer from substantial batch effects, which often lead to poor generalization between the sets despite the informative (common) internal structure of each set individually.

To that end, we present several methods for Domain Adaptation (DA) on Riemannian manifolds. Specifically, we first show how Parallel Transport (PT) on the manifold of Symmetric Positive Definite (SPD) matrices can be used for DA. We then extend this work to the manifold of Symmetric Positive Semi-Definite (SPSD) matrices (and to the Grassmann manifold). In addition, we propose and analyze a statistical approach for DA on the manifold of SPD matrices using Optimal Transport (OT). Using the polar factorization theorem, we show that although OT for DA is very useful, it suffers from a fundamental limitation, since it cannot recover volume preserving transformations.


In the second part we consider the problem of common feature extraction from multimodal datasets.

When the obtained observation sets are acquired from different models, DA is ineffective. Instead, we can extract common features from the given sets. We approach this task using manifold learning.

We propose to construct a local Riemannian metric for sensor data fusion, which can be viewed as a local Canonical Correlation Analysis (CCA), which, in turn, can be incorporated in a kernel-based manifold learning technique.


In addition, we propose a new spectral method for constructing functions that are jointly smooth on multiple observed manifolds. This method is unsupervised and primarily consists of two steps. First, using kernels, we obtain a subspace spanning smooth functions on each manifold. Then, we apply a spectral method to the obtained subspaces and extract functions that are jointly smooth on all manifolds. We show analytically that our method is guaranteed to provide a set of orthogonal functions that are as jointly smooth as possible, ordered from the smoothest to the least smooth.


In the third part we consider the problem of non-linear dynamical system analysis based on observations. Creating minimal descriptions of parametrically-dependent unknown non-linear dynamical systems is a long-standing problem and is considered challenging, especially in a model-free setting. Using an implementation of data-informed geometry learning, we directly reconstruct a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Specifically, we show how metric estimation and manifold learning can used to provide non-parametric representation for dynamical systems and identify different dynamical regimes, bifurcations, and phase transitions solely from observations.