Generalized designs on graphs: Sampling, spectra, symmetries

Abstract

Spherical designs are finite sets of points on the sphere S d with the property that the average of certain (low‐degree) polynomials in these points coincides with the global average of the polynomial on S d. They are evenly distributed and often exhibit a great degree of regularity and symmetry. We point out that a spectral definition of spherical designs transfers to finite graphs—these ‘graphical designs’ are subsets of vertices that are evenly spaced and capture the symmetries of the underlying graph (should they exist). Our main result states that good graphical designs either consist of many vertices or their neighborhoods have exponential volume growth. We show several examples, describe ways to find them and discuss problems.

Numerical integration on graphs: Where to sample and how to weigh

“…Indeed, it is a fairly common technique to design quadrature rules for a certain class of very smooth functions -they will usually still be somewhat effective on functions that are not as smooth as the functions used in the design of the rule. This is, of course, the main idea in the construction of spherical designs [6,14,16] but also appears in other contexts: the Simpson rule for the integration of a real function on an interval is designed to be exact on quadratic polynomials.…”

“…The purpose of this paper is to report on a very general idea in the context of sampling on graphs. It is a companion paper to [16] dealing with a problem in Spectral Graph Theory. We extend some of the ideas from [15,16] to sampling, prove an inequality bounding the integration error in terms of the geometry of the sampling points and give several examples.…”

“…It is a companion paper to [16] dealing with a problem in Spectral Graph Theory. We extend some of the ideas from [15,16] to sampling, prove an inequality bounding the integration error in terms of the geometry of the sampling points and give several examples. We will, throughout the paper, use G = (V, E, w) to denote a connected graph with weighted edges.…”

Quadrature Points via Heat Kernel Repulsion

“…We emphasize, however, that these spherical design point sets are built so as to minimize the error functional, are generally very hard to construct, and may only exist for certain values of N . For general smooth manifolds the explicit construction of point sets which exactly integrate a prescribed number of Laplacian eigenfunctions is a not well understood problem (analogous notions of designs seem to exist in a fairly abstract setting, see [52] and [47], but we are not aware of any reliable way they could be constructed on manifolds that are not spheres). 3.3.…”

“…We refer to [50] for a recent refinement of Montgomery's result and to recent work of Bilyk, Dai and the third author [5] for general refinements. The paper [51] establishes that a set of N points on a compact d−dimensional manifold cannot integrate more than the first c d N + o(N ) eigenfunctions exactly (see also [2,52]). Both papers [5,51] make use of the heat kernel.…”