Positive definite matrix-valued kernels and their scalar valued projections: counterexamples

TL;DR

This paper demonstrates that certain positive definite matrix-valued kernels do not behave well under scalar projections, providing counterexamples and generalizations in an abstract setting involving invariant kernels.

Contribution

It introduces counterexamples showing the failure of scalar projections to preserve positive definiteness in matrix-valued kernels and generalizes these results using invariant kernel concepts.

Findings

Counterexamples for positive definite matrix-valued kernels

Scalar projections may not preserve positive definiteness

Generalization to invariant kernel settings

Abstract

In this paper we show that the strictly positive definite matrix valued isotropic kernels in the circle and the real dot product kernels in Euclidean spaces are not well behaved with respect to its scalar valued projections. We generalize the counterexamples that we obtained to an abstract setting by using the concepts of unitarily invariant kernels and adjointly invariant kernels, provided the existence of an aperiodic invariant function.