More precisely, we consider networks with a single hidden layer,obtained bysumming channels formed byapplying anequivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linearoutputlayer.

Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or \emph{equivariant} (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linear output layer. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions. Our main contribution is then an extension of this result to the \emph{equivariant} case, which appears in many practical applications but has been less studied from a theoretical point of view. The proof relies on a new generalized Stone-Weierstrass theorem for algebra of equivariant functions, which is of independent interest. Additionally, unlike many previous works that consider a fixed number of nodes, our results show that a GNN defined by a single set of parameters can approximate uniformly well a function defined on graphs of varying size.

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone--Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.

Recently, Maron et al. (2019b) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained.

Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or \emph{equivariant} (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linear output layer. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions.

Transformers are deep architectures that define "in-context mappings" which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for vision transformers). This work studies in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically and uniformly address the expressivity of these architectures, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens (discrete for a finite number of tokens). The related notion of smoothness corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLP layers between multi-head attention layers is also explicitly controlled.

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result for generalizations of continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and show that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves the way towards uncertainty quantification for signature kernel regression.

In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some family of graphs (e.g. paths or non-isomorphic trees). We show that graph homomorphism numbers provide a natural invariant (isomorphism invariant and $\mathcal{F}$-invariant) embedding maps which can be used for graph classification. Viewing the expressive power of a graph classifier by the $\mathcal{F}$-indistinguishable concept, we prove the universality property of graph homomorphism vectors in approximating $\mathcal{F}$-invariant functions. In practice, by choosing $\mathcal{F}$ whose elements have bounded tree-width, we show that the homomorphism method is efficient compared with other methods.

Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or \emph{equivariant} (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linear output layer. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions.

We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality conclusions including an open case for higher-order output. We find that our proposed technique not only leads to simple proofs of the universality properties but also gives a natural explanation for the tensorization of the previously studied models. Finally, we give some remarks on the connection between our model and the continuous representation of graphs.