Data-Driven Discovery of Linear Dynamical Systems Over Graphs via Dynamical Sampling

Explore data-driven discovery of linear dynamical systems over graphs through dynamical sampling in this lecture from the Focus Program on Analytic Function Spaces and their Applications. Delve into graph signal processing, examining how to recover missing information on graph nodes efficiently and robustly. Investigate preliminaries of graph signal sampling, its relation to frame/basis theory in harmonic analysis, and previous work on deterministic dynamical sampling. Learn about space-time sampling formulation and three proposed random space-time sampling regimes. Discover optimal sampling distributions based on graph structure and reconstruction strategy. Examine guarantees for standard and efficient decoders, system identification in dynamical sampling, and generalizations to affine systems. Conclude with numerical results demonstrating the effectiveness of the presented methods.

Intro
Graph signal processing Given information at a subset nodes of a graph, can we recover the missing information on other modes in a robust and efficient way?
Preliminaries
Sampling of graph signals
Motivation
Relation to Frame/Basis theory in Harmonic analysis
Previous work on deterministic dynamical sampling
Relation to linear inverse problem How to choose space-time samples that can do as good as spatial samples? Formulation We can write the space-time sampling as
Randomized dynamical sampling We propose three different random space-time sampling regimes
Random space-time sampling model
Connection with the static case T=1
Optimal sampling distributions
Summary • Optimal sampling distribution depends on the graph structure and the
Reconstruction strategy
Guarantees for standard decoder
Guarantees for efficient decoder
System Identification in dynamical sampling
Generalization to affine systems
Numerical Results