W375 Westgate Building
10:00AM
ABSTRACT
Spectral methods such as principal component analysis and canonical correlation analysis are foundational tools in statistics for extracting low-dimensional structure from high-dimensional data, and their streaming and stochastic variants play a central role in modern large-scale inference. At the same time, many emerging problems in science and engineering require estimating the spectral structure of infinite-dimensional operators on function spaces over high-dimensional domains, including differential operators and conditional expectation operators. In these regimes, classical Rayleigh-quotient-based formulations, which rely on explicit orthogonality constraints, often become unstable and difficult to scale, particularly in parametric or learned settings.
In this talk, I present a reformulation of spectral estimation based on unconstrained variational objectives that implicitly encode ordered spectral structure through nested optimization. This approach removes the need for explicit orthogonality constraints while preserving spectral ordering, and is naturally compatible with large-scale stochastic optimization. The resulting framework provides a unified and computationally efficient approach to learning structured representations in high-dimensional problems, without requiring explicit orthogonalization or projection steps.
I illustrate this framework in three domains: eigenvalue problems for differential operators such as the Schrödinger equation in quantum chemistry, spectral (Koopman) analysis of nonlinear dynamical systems including molecular dynamics, and structured representation learning with deep neural networks. Together, these examples demonstrate how principled reformulations of classical spectral objectives can enable scalable, reliable AI methods for scientific discovery.
Additional Information:
BIOGRAPHY
Jongha (Jon) Ryu is a postdoctoral associate in the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. He received his Ph.D. in Electrical and Computer Engineering from the University of California San Diego. His research develops statistical and mathematical foundations for artificial intelligence, with a focus on spectral learning, generative modeling, and uncertainty quantification for scientific and engineering systems.
Details...
