On the approximation capability of GNNs in node classification/regression tasks

Graph neural networks (GNNs) are a broad class of connectionist models for graph processing. Recent studies have shown that GNNs can approximate any function on graphs, modulo the equivalence relation on graphs defined by theWeisfeiler-Lehman (WL) test. However, these results suffer from some limitations, both because they were derived using the Stone-Weierstrass theorem — which is existential in nature — and because they assume that the target function to be approximated must be continuous. Furthermore, all current results are dedicated to graph classification/regression tasks, where the GNN must produce a single output for the whole graph, while also node classification/regression problems, in which an output is returned for each node, are very common. In this paper, we propose an alternative way to demonstrate the approximation capability of GNNs that overcomes these limitations. Indeed, we show that GNNs are universal approximators in probability for node classification/regression tasks, as they can approximate any measurable function that satisfies the 1-WL-equivalence on nodes. The proposed theoretical framework allows the approximation of generic discontinuous target functions and also suggests the GNN architecture that can reach a desired approximation. In addition, we provide a bound on the number of the GNN layers required to achieve the desired degree of approximation, namely 2r − 1, where r is the maximum number of nodes for the graphs in the domain

A collocation IGA-BEM for 3D potential problems on unbounded domains

In this paper the numerical solution of potential problems defined on 3D unbounded domains is addressed with Boundary Element Methods (BEMs), since in this way the problem is studied only on the boundary, and thus any finite approximation of the infinite domain can be avoided. The isogeometric analysis (IGA) setting is considered and in particular B-splines and NURBS functions are taken into account. In order to exploit all the possible benefits from using spline spaces, an important point is the development of specific cubature formulas for weakly and nearly singular integrals. Our proposal for this aim is based on spline quasi-interpolation and on the use of a spline product formula. Besides that, a robust singularity extraction procedure is introduced as a preliminary step and an efficient function-by-function assembly phase is adopted. A selection of numerical examples confirms that the numerical solutions reach the expected convergence orders.Comment: 17 pages, 4 figure

Effective scheme for constrained curve construction

Dottorato di ricerca in matematica computazionale e ricerca operativa. 10 cicloConsiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome; Biblioteca Nazionale Centrale - P.za Cavalleggeri, 1, Florence / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Constrained Interpolation in R^3 by Abstract Schemes

In this paper we present abstract schemes for the construction of shape preserving interpolating space curves. The proposed method uses piecewise defined curves whose pieces are taken from standard low degree polynomial spaces without tension properties. Moreover with this approach many optimization functionals (for instance any fairness functional) can be used to select the best interpolant among the admissible ones

Support Function Representation for Curvature Dependent Surface Sampling

In many applications it is required to have a curvature-dependent surface sampling, based on a local shape analysis. In this work we show how this can be achieved by using the support function (SF) representation of a surface. This representation, a classical tool in Convex Geometry, has been recently considered in CAD problems for computing surface offsets and for analyzing curvatures. Starting from the observation that triangular Bézier spline surfaces have quite simple support functions, we approximate any given free-form surface by a quadratic triangular Bézier spline surface. Then the corresponding approximate SF representation can be efficiently exploited to produce a curvature dependent sampling of the approximated surface

Abstract Schemes and Constrained Curve Interpolation

The purpose of this paper is to describe how a method similar to alternating projections can be used to obtain viable algorithms for a general and abstract formulation of constrained curve construction problem

Geometric interpolation of ER frames with G2 Pythagorean-hodograph curves of degree 7

The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial extcolor{blue}{PH} curve, we consider the Euler-Rodrigues (ER) frame. Moreover G^1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of G^k continuity of ER frames implies G^{k+1} continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results

A unifying point of view on expressive power of GNNs

Graph Neural Networks (GNNs) are a wide class of connectionist models for graph processing. They perform an iterative message passing operation on each node and its neighbors, to solve classification/ clustering tasks -- on some nodes or on the whole graph -- collecting all such messages, regardless of their order. Despite the differences among the various models belonging to this class, most of them adopt the same computation scheme, based on a local aggregation mechanism and, intuitively, the local computation framework is mainly responsible for the expressive power of GNNs. In this paper, we prove that the Weisfeiler--Lehman test induces an equivalence relationship on the graph nodes that exactly corresponds to the unfolding equivalence, defined on the original GNN model. Therefore, the results on the expressive power of the original GNNs can be extended to general GNNs which, under mild conditions, can be proved capable of approximating, in probability and up to any precision, any function on graphs that respects the unfolding equivalence.Comment: 16 pages, 3 figure

Spline surfaces with C1 quintic PH isoparametric curves

Given two spatial PH spline curves, aim of this paper is to study the construction of a tensor–product spline surface which has the two curves as assigned boundaries and which in addition incorporates a single family of isoparametric PH spline curves. Such a construction is carried over in two steps. In the first step a bi–patch is determined in a ‘Coons–like’ way having as boundaries two quintic PH curves forming a single section of given spline curves, and two polynomial quartic curves. In the second step the bi–patches are put together to form a globally C1 continuous surface. In order to determine the final shape of the resulting surface, some free parameters are set by minimizing suitable shape functionals. The method can be extended to general boundary curves by preliminary approximating them with quintic PH splines