Why not to use the Gaussian kernel

A new paper submitted to arXiv on August 27, 2026, argues that the Gaussian kernel, also known as the squared exponential or radial basis function kernel, should never be used as a default in Gaussian process regression because it is extremely brittle, leading to unrealistically small conditional variance and numerical ill-conditioning. The authors contend that analytic kernels in general are best avoided due to their unnatural smoothness.

Statistics Machine Learning Submitted on 27 Aug 2026 Title:Why not to use the Gaussian kernel View PDF /pdf/2608.26974 Abstract:Kernels measure similarity or correlation in tasks such as regression and classification. The Gaussian kernel, other names of which include squared exponential and radial basis function kernel, is one of the most popular in Gaussian process regression. We argue that the Gaussian kernel is best avoided and should never be used as a default. The argument rests on two results demonstrating that the Gaussian kernel is extremely brittle. First, the Gaussian kernel gives rise to a conditional variance that is unrealistically small. If the variance is used to quantify predictive uncertainty, catastrophic overconfidence is almost inevitable. Second, a small variance goes hand in hand with numerical ill-conditioning, so that to use the Gaussian kernel in practice requires tricks such as nugget terms that effectively modify the underlying regression or classification model. These problems are caused by the unnatural smoothness of the Gaussian kernel, a fact we are far from the first to take notice of. The problem is not the Gaussian form itself but the analyticity of the kernel: Our argument is more broadly that analytic kernels are best avoided. For stationary kernels analyticity is essentially equivalent to an exponential decay of the spectral density. Current browse context: stat.ML References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .