Core Concepts
Efficiently creating locally supported basis functions on graphs.
Abstract
The article discusses the creation of locally supported basis functions on graphs for efficient function approximation. It proposes perturbations of Lagrange bases on graphs with local support, derived from a differential operator. The study includes error estimates between local and interpolatory Lagrange basis functions, demonstrating their utility in numerical experiments. The paper focuses on constructing bases efficiently using only local information, leading to benefits like parallel computation and sparse data storage. The analysis is based on adjacency and Laplacian matrices, considering standard, normalized, and random-walk variants.
Stats
Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate ∥A∥∞ ≤ √n ∥A∥2.
We show that with reasonable assumptions on the graph, these proposed functions can be made arbitrarily close to the full Lagrange functions.