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On Euclidean spaces, the Gaussian kernel is one of the most widely usedkernels in applications. It has also been used on non-Euclidean spaces, whereit is known that there may be (and often are) scale parameters for which it isnot positive definite. Hope remains that this kernel is positive definite formany choices of parameter. However, we show that the Gaussian kernel is notpositive definite on the circle for any choice of parameter. This implies thaton metric spaces in which the circle can be isometrically embedded, such asspheres, projective spaces and Grassmannians, the Gaussian kernel is notpositive definite for any parameter.

The Gaussian kernel on the circle and spaces that admit isometric embeddings of the circle