The optimal assignment kernel is not positive definite

TL;DR

This paper demonstrates that the optimal assignment kernel, used for embedding labeled graphs into Hilbert spaces, does not always satisfy the positive definiteness property, challenging its theoretical foundation.

Contribution

It provides a proof that the optimal assignment kernel is not universally positive definite, clarifying limitations of this embedding approach.

Findings

Optimal assignment kernel is not always positive definite.

Challenges assumptions about kernel validity in graph embedding.

Implications for graph kernel methods and their theoretical guarantees.

Abstract

We prove that the optimal assignment kernel, proposed recently as an attempt to embed labeled graphs and more generally tuples of basic data to a Hilbert space, is in fact not always positive definite.