Applied Mathematics Colloquia by Houman Owhadi - Co-Discovering Graphical Structure and Functional Relationships Within Data: A Gaussian Process Framework for Connecting the Dots

Speaker: Houman Owhadi, professor of applied and computational mathematics and dynamical systems, California Institute of Technology

Title: Co-Discovering Graphical Structure and Functional Relationships Within Data: A Gaussian Process Framework for Connecting the Dots

Abstract: Most scientific challenges can be framed into one of the following three levels of complexity of function approximation. 

• Type 1: Approximate an unknown function given input/output data. 
• Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. 
• Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. 

Examples of Type 2 problems include solving and learning (possibly stochastic) nonlinear partial differential equations (PDEs), while Type 3 problems encompass learning dependencies between variables in a mechanical system, identifying chemical reaction networks, and determining relationships between genes through a protein-signaling network. Although Gaussian Process (GP) methods are sometimes perceived as a well-founded but old technology limited to Type 1 curve fitting, they can be generalized to an interpretable framework for solving Type 2 and Type 3 problems, all while maintaining the simple and transparent theoretical and computational guarantees of kernel/optimal recovery methods.

Applied Mathematics Colloquium