On the Hölder Stability of Multiset and Graph Neural Networks

Extensive research efforts have been put into characterizing and constructing maximally separating multiset and graph neural networks. However, recent empirical evidence suggests the notion of separation itself doesn't capture several interesting phenomena. On the one hand, the quality of this separation may be very weak, to the extent that the embeddings of "separable" objects might even be considered identical when using fixed finite precision. On the other hand, architectures which aren't capable of separation in theory, somehow achieve separation when taking the network to be wide enough.