arXiv subjects

Floris Geerts

Publications and source records attributed to Floris Geerts.

34 records · Page 2Linked to original sources

arXiv2022

Expressiveness and Approximation Properties of Graph Neural Networks

Characterizing the separation power of graph neural networks (GNNs) provides an understanding of their limitations for graph learning tasks. Results regarding separation power are, however, usually geared at specific GNN architectures, and tools for understanding arbitrary GNN architectures are generally lacking. We provide an elegant way to easily obtain bounds on the separation power of GNNs in terms of the Weisfeiler-Leman (WL) tests, which have become the yardstick to measure the separation power of GNNs. The crux is to view GNNs as expressions in a procedural tensor language describing the computations in the layers of the GNNs. Then, by a simple analysis of the obtained expressions, in terms of the number of indexes and the nesting depth of summations, bounds on the separation power in terms of the WL-tests readily follow. We use tensor language to define Higher-Order Message-Passing Neural Networks (or k-MPNNs), a natural extension of MPNNs. Furthermore, the tensor language point of view allows for the derivation of universality results for classes of GNNs in a natural way. Our approach provides a toolbox with which GNN architecture designers can analyze the separation power of their GNNs, without needing to know the intricacies of the WL-tests. We also provide insights in what is needed to boost the separation power of GNNs.

cs.LG↗

arXiv2022

On the expressive power of message-passing neural networks as global feature map transformers

We investigate the power of message-passing neural networks (MPNNs) in their capacity to transform the numerical features stored in the nodes of their input graphs. Our focus is on global expressive power, uniformly over all input graphs, or over graphs of bounded degree with features from a bounded domain. Accordingly, we introduce the notion of a global feature map transformer (GFMT). As a yardstick for expressiveness, we use a basic language for GFMTs, which we call MPLang. Every MPNN can be expressed in MPLang, and our results clarify to which extent the converse inclusion holds. We consider exact versus approximate expressiveness; the use of arbitrary activation functions; and the case where only the ReLU activation function is allowed.

cs.AI↗

arXiv2021

Graph Neural Networks with Local Graph Parameters

Various recent proposals increase the distinguishing power of Graph Neural Networks GNNs by propagating features between $k$-tuples of vertices. The distinguishing power of these "higher-order'' GNNs is known to be bounded by the $k$-dimensional Weisfeiler-Leman (WL) test, yet their $\mathcal O(n^k)$ memory requirements limit their applicability. Other proposals infuse GNNs with local higher-order graph structural information from the start, hereby inheriting the desirable $\mathcal O(n)$ memory requirement from GNNs at the cost of a one-time, possibly non-linear, preprocessing step. We propose local graph parameter enabled GNNs as a framework for studying the latter kind of approaches and precisely characterize their distinguishing power, in terms of a variant of the WL test, and in terms of the graph structural properties that they can take into account. Local graph parameters can be added to any GNN architecture, and are cheap to compute. In terms of expressive power, our proposal lies in the middle of GNNs and their higher-order counterparts. Further, we propose several techniques to aide in choosing the right local graph parameters. Our results connect GNNs with deep results in finite model theory and finite variable logics. Our experimental evaluation shows that adding local graph parameters often has a positive effect for a variety of GNNs, datasets and graph learning tasks.

cs.LG↗

arXiv2020

Expressive power of linear algebra query languages

Linear algebra algorithms often require some sort of iteration or recursion as is illustrated by standard algorithms for Gaussian elimination, matrix inversion, and transitive closure. A key characteristic shared by these algorithms is that they allow looping for a number of steps that is bounded by the matrix dimension. In this paper we extend the matrix query language MATLANG with this type of recursion, and show that this suffices to express classical linear algebra algorithms. We study the expressive power of this language and show that it naturally corresponds to arithmetic circuit families, which are often said to capture linear algebra. Furthermore, we analyze several sub-fragments of our language, and show that their expressive power is closely tied to logical formalisms on semiring-annotated relations.

cs.CC↗

arXiv2020

The expressive power of kth-order invariant graph networks

The expressive power of graph neural network formalisms is commonly measured by their ability to distinguish graphs. For many formalisms, the k-dimensional Weisfeiler-Leman (k-WL) graph isomorphism test is used as a yardstick. In this paper we consider the expressive power of kth-order invariant (linear) graph networks (k-IGNs). It is known that k-IGNs are expressive enough to simulate k-WL. This means that for any two graphs that can be distinguished by k-WL, one can find a k-IGN which also distinguishes those graphs. The question remains whether k-IGNs can distinguish more graphs than k-WL. This was recently shown to be false for k=2. Here, we generalise this result to arbitrary k. In other words, we show that k-IGNs are bounded in expressive power by k-WL. This implies that k-IGNs and k-WL are equally powerful in distinguishing graphs.

cs.LG↗

arXiv2020

Walk Message Passing Neural Networks and Second-Order Graph Neural Networks

The expressive power of message passing neural networks (MPNNs) is known to match the expressive power of the 1-dimensional Weisfeiler-Leman graph (1-WL) isomorphism test. To boost the expressive power of MPNNs, a number of graph neural network architectures have recently been proposed based on higher-dimensional Weisfeiler-Leman tests. In this paper we consider the two-dimensional (2-WL) test and introduce a new type of MPNNs, referred to as $\ell$-walk MPNNs, which aggregate features along walks of length $\ell$ between vertices. We show that $2$-walk MPNNs match 2-WL in expressive power. More generally, $\ell$-walk MPNNs, for any $\ell\geq 2$, are shown to match the expressive power of the recently introduced $\ell$-walk refinement procedure (W[$\ell$]). Based on a correspondence between 2-WL and W[$\ell$], we observe that $\ell$-walk MPNNs and $2$-walk MPNNs have the same expressive power, i.e., they can distinguish the same pairs of graphs, but $\ell$-walk MPNNs can possibly distinguish pairs of graphs faster than $2$-walk MPNNs. When it comes to concrete learnable graph neural network (GNN) formalisms that match 2-WL or W[$\ell$] in expressive power, we consider second-order graph neural networks that allow for non-linear layers. In particular, to match W[$\ell$] in expressive power, we allow $\ell-1$ matrix multiplications in each layer. We propose different versions of second-order GNNs depending on the type of features (i.e., coming from a countable set, or coming from an uncountable set) as this affects the number of dimensions needed to represent the features. Our results indicate that increasing non-linearity in layers by means of allowing multiple matrix multiplications does not increase expressive power. At the very best, it results in a faster distinction of input graphs.

cs.LG↗

arXiv2020

Let's Agree to Degree: Comparing Graph Convolutional Networks in the Message-Passing Framework

In this paper we cast neural networks defined on graphs as message-passing neural networks (MPNNs) in order to study the distinguishing power of different classes of such models. We are interested in whether certain architectures are able to tell vertices apart based on the feature labels given as input with the graph. We consider two variants of MPNNS: anonymous MPNNs whose message functions depend only on the labels of vertices involved; and degree-aware MPNNs in which message functions can additionally use information regarding the degree of vertices. The former class covers a popular formalisms for computing functions on graphs: graph neural networks (GNN). The latter covers the so-called graph convolutional networks (GCNs), a recently introduced variant of GNNs by Kipf and Welling. We obtain lower and upper bounds on the distinguishing power of MPNNs in terms of the distinguishing power of the Weisfeiler-Lehman (WL) algorithm. Our results imply that (i) the distinguishing power of GCNs is bounded by the WL algorithm, but that they are one step ahead; (ii) the WL algorithm cannot be simulated by "plain vanilla" GCNs but the addition of a trade-off parameter between features of the vertex and those of its neighbours (as proposed by Kipf and Welling themselves) resolves this problem.

cs.LG↗

arXiv2020

On the expressive power of linear algebra on graphs

Most graph query languages are rooted in logic. By contrast, in this paper we consider graph query languages rooted in linear algebra. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. A complete picture is painted of the impact of the linear algebra operations in MATLANG on their ability to distinguish graphs.

cs.DB↗

arXiv2018

A General Framework for Anytime Approximation in Probabilistic Databases

Anytime approximation algorithms that compute the probabilities of queries over probabilistic databases can be of great use to statistical learning tasks. Those approaches have been based so far on either (i) sampling or (ii) branch-and-bound with model-based bounds. We present here a more general branch-and-bound framework that extends the possible bounds by using 'dissociation', which yields tighter bounds.

cs.DB↗

arXiv2017

On the expressive power of query languages for matrices

We investigate the expressive power of $\mathsf{MATLANG}$, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation $\mathsf{inv}$ of inverting a matrix. In $\mathsf{MATLANG}+\mathsf{inv}$ we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We also consider an operation $\mathsf{eigen}$ for diagonalizing a matrix, which is defined so that different eigenvectors returned for a same eigenvalue are orthogonal. We show that $\mathsf{inv}$ can be expressed in $\mathsf{MATLANG}+\mathsf{eigen}$. We put forward the open question whether there are boolean queries about matrices, or generic queries about graphs, expressible in $\mathsf{MATLANG} + \mathsf{eigen}$ but not in $\mathsf{MATLANG}+\mathsf{inv}$. The evaluation problem for $\mathsf{MATLANG} + \mathsf{eigen}$ is shown to be complete for the complexity class $\exists \mathbf{R}$.

cs.DB↗

arXiv2006

On-line topological simplification of weighted graphs

We describe two efficient on-line algorithms to simplify weighted graphs by eliminating degree-two vertices. Our algorithms are on-line in that they react to updates on the data, keeping the simplification up-to-date. The supported updates are insertions of vertices and edges; hence, our algorithms are partially dynamic. We provide both analytical and empirical evaluations of the efficiency of our approaches. Specifically, we prove an O(log n) upper bound on the amortized time complexity of our maintenance algorithms, with n the number of insertions.

cs.DS↗

arXiv2005

First-order Complete and Computationally Complete Query Languages for Spatio-Temporal Databases

We address a fundamental question concerning spatio-temporal database systems: ``What are exactly spatio-temporal queries?'' We define spatio-temporal queries to be computable mappings that are also generic, meaning that the result of a query may only depend to a limited extent on the actual internal representation of the spatio-temporal data. Genericity is defined as invariance under groups of geometric transformations that preserve certain characteristics of spatio-temporal data (e.g., collinearity, distance, velocity, acceleration, ...). These groups depend on the notions that are relevant in particular spatio-temporal database applications. These transformations also have the distinctive property that they respect the monotone and unidirectional nature of time. We investigate different genericity classes with respect to the constraint database model for spatio-temporal databases and we identify sound and complete languages for the first-order and the computable queries in these genericity classes. We distinguish between genericity determined by time-invariant transformations, genericity notions concerning physical quantities and genericity determined by time-dependent transformations.

cs.DB↗

arXiv2004

Frequent Knot Discovery

We explore the possibility of applying the framework of frequent pattern mining to a class of continuous objects appearing in nature, namely knots. We introduce the frequent knot mining problem and present a solution. The key observation is that a database consisting of knots can be transformed into a transactional database. This observation is based on the Prime Decomposition Theorem of knots.

cs.DB↗

arXiv2003

Two- versus three-dimensional connectivity testing of first-order queries to semi-algebraic sets

This paper addresses the question whether one can determine the connectivity of a semi-algebraic set in three dimensions by testing the connectivity of a finite number of two-dimensional ``samples'' of the set, where these samples are defined by first-order queries. The question is answered negatively for two classes of first-order queries: cartesian-product-free, and positive one-pass.

cs.LO↗

arXiv2002

A Tight Upper Bound on the Number of Candidate Patterns

In the context of mining for frequent patterns using the standard levelwise algorithm, the following question arises: given the current level and the current set of frequent patterns, what is the maximal number of candidate patterns that can be generated on the next level? We answer this question by providing a tight upper bound, derived from a combinatorial result from the sixties by Kruskal and Katona. Our result is useful to reduce the number of database scans.

cs.DB↗

arXiv2001

Expressing the cone radius in the relational calculus with real polynomial constraints

We show that there is a query expressible in first-order logic over the reals that returns, on any given semi-algebraic set A, for every point a radius around which A is conical. We obtain this result by combining famous results from calculus and real algebraic geometry, notably Sard's theorem and Thom's first isotopy lemma, with recent algorithmic results by Rannou.

cs.DB↗