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Pixel Club: Recovering Dynamics in Geometry Processing, Fluid Mechanics, Image Processing and Social Sciences via Koopman Theory
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Omri Azencot (University of California, LA)
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Tuesday, 7.1.2020, 11:30
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Electrical Eng. Building 1061
Dynamical systems are everywhere, from the flow of particles in the air to the evolution of people's personalities. Recently, the big data revolution has made increasing amounts of observations of such systems readily available. Unfortunately, in many practical scenarios, the governing equations of the dynamics are unknown. For instance, what would be the mathematical model for periodic questionnaire data that describes personalities? Despite the lack of existing models, scientists and engineers are required to address challenging problems with these type of data on a daily basis.

In this talk, we will present the main building blocks that allow for an interpretative analysis and process of dynamical systems data. The overarching theme of our work is based on the theory of Bernard Koopman (1931). Key to our approach is the interplay between the underlying dynamics and its embedding onto an infinite-dimensional space of scalar functions. This embedding encodes a potentially nonlinear system via a linear object known as the Koopman Operator. Koopman's perspective is advantageous since it involves the manipulation of linear operators which can be done efficiently. Moreover, algebraic properties of the Koopman matrix are directly associated with dynamical features of the system, which in turn are linked to high-level questions. Overall, the combination of Koopman Theory with novel dimensionality reduction techniques and data science approaches leads to a highly powerful framework.

To demonstrate the effectiveness of our approach, we consider several challenging problems in various fields. In geometry processing, we show how optimizing for a spectral basis and a Koopman operator (a functional map) leads to improved shape matching results. In fluid mechanics, we develop a provably convergent ADMM scheme for computing Koopman operators that admits state-of-the-art results on data with high levels of sensor noise. In image processing, our methodology generates a discrete transform of a nonlinear flow as faster as two orders of magnitude when compared to existing approaches. Finally, we construct a novel framework for recovering dynamics from questionnaire data that arise in the social sciences.

We will conclude the talk by discussing current and future work. In particular, we are actively working on designing a variational autoencoder that maintains temporal relations while allowing for the inference of a Koopman Operator. Lastly, we also consider the problem of fully recovering the governing equations from a given dataset or a Koopman matrix.

Omri Azencot is an Assistant Adjunct Professor in the Department of Mathematics at The University of California, Los Angeles. His research interests include geometry processing, dynamical systems and optimization. In particular, Omri is interested in designing theoretically sound and effective algorithms for challenging problems such as shape matching, recovering fluid flows, and designing nonlinear image transforms. Omri completed his PhD at the Computer Science Department at the Technion -- Israel Institute of Technology, under the supervision of Prof. Mirela Ben-Chen. Prior to that, he obtained a BSc in computer science and a BSc in mathematics, both from the Technion. Omri's research has been supported by an Adams Fellowship of the Israel Academy of Sciences and Humanities, a Zuckerman Postdoctoral Fellowship, and a Marie Sklodowska-Curie Actions International Fellowship.
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