Extrapolation and sampling for processes on spatial graphs

TL;DR

This paper introduces a spectral framework for processes on spatial graphs, providing conditions for unique extrapolation and sampling-based recovery, extending classical sampling theory to complex graph-structured domains.

Contribution

It develops a spectral degeneracy concept considering graph topology and establishes conditions for unique extrapolation and sampling recovery on spatial graphs.

Findings

Spectral degeneracy notion tailored to graph topology.

Conditions for unique extrapolation from single-branch observations.

Sampling criteria for process recovery from equidistant samples.

Abstract

The paper studies processes defined on time domains structured as oriented spatial graphs (or metric graphs, or oriented branched 1-manifolds). This setting can be used, for example, for forecasting models involving branching scenarios. For these processes, a notion of the spectrum degeneracy that takes into account the topology of the graph is introduced. The paper suggests sufficient conditions of uniqueness of extrapolation and recovery from the observations on a single branch. This also implies an analog of sampling theorem for branching processes, i.e., criterions of their recovery from a set of equidistant samples, as well as from a set of equidistant samples from a single branch.