Graduate Students Seminar

Session Chair:   Fred Azizi
Discussant:        Dr. Kvalheim

Speaker 1:  Pratyusha Sarkar 

Title: Discovering Subject-Specific Dynamical Systems with Varying Coefficients via Local Polynomial Regression 

Abstract
Growth processes in the natural and biomedical sciences are often governed by dynamical systems. The parameters of this dynamical system vary across individuals, an easy example of this phenomena can be observPratyushaed in personalized medicine. We propose a locally weighted polynomial regression framework for discovering subject-specific dynamical systems from sparse, noisy observations. We develop two strategies, a sequential approach where we first fit the local regression and use its results to obtain the dynamical system, and a simultaneous approach, where we estimate both the local regression and dynamical system simultaneously along with a principled bandwidth-selection strategy that accounts for their cascading dependency. We embed the dynamics in a varying-coefficient specification, allowing the growth rate to depend flexibly on subject-level covariates. The framework extends to multi-subject settings via random effects capturing heterogeneity in initial conditions and growth rates. Simulations demonstrate accurate recovery and coverage across signal-to-noise and sampling regimes.

Speaker 2: Josh McCarter  

Title: Inferring Dynamical Properties from Latent Models is Problematic

Abstract
Dynamical systems arising in nature and in engineering applications are often high-dimensional, yet they may admit much lower-dimensional “latent” representations. Due to the high-dimensionality of the original system, study of its dynamical properties is typically both analytically and computationally prohibitive, so one is often forced to utilize the latent representation for these studies. Data-driven methods, particularly autoencoder architectures, are commonly used to discover such latent representations. But how can we know whether representations discovered by autoencoders meaningfully capture dynamical information of the original system? In this talk, we present several new results showing that an extremely simple latent dynamical system is capable of representing a broad class of high-dimensional dynamical systems on an invariant manifold of a given dimension. In this sense, our results provide a negative answer to the question posed above: the existence of a low-dimensional representation does not, by itself, guarantee that the representation captures the essential dynamical structure of the original system.