Scholarlink

<p>Gaussian Process regression is a kernel method successfully adopted in many real-life applications. Recently, there is a growing interest on extending this method to non-Euclidean input spaces, like the one considered in this paper, consisting of probability measures. Although a Positive Definite kernel can be defined by using a suitable distance—the Wasserstein distance— the common procedure for learning the Gaussian Process model can fail due to numerical issues, arising earlier and more frequently than in the case of an Euclidean input space and, as demonstrated, impossible to avoid by adding artificial noise (<i>nugget effect</i>) as usually done. This paper uncovers the main reason of these issues, that is a non-stationarity relation between the Wasserstein-based squared exponential kernel and its Euclidean counterpart. As a relevant result, we learn a Gaussian Process model by assuming the input space as Euclidean and then use an algebraic transformation, based on the uncovered relation, to transform it into a non-stationary and Wasserstein-based Gaussian Process model over probability measures. This algebraic transformation is simpler than <i>log-exp</i> maps used on data belonging to Riemannian manifolds and recently extended to consider the pseudo-Riemannian structure of an input space equipped with the Wasserstein distance.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernels

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年1月11日

查看全文
  • Antonio Candelieri,
  • Andrea Ponti,
  • Francesco Archetti

摘要

Gaussian Process regression is a kernel method successfully adopted in many real-life applications. Recently, there is a growing interest on extending this method to non-Euclidean input spaces, like the one considered in this paper, consisting of probability measures. Although a Positive Definite kernel can be defined by using a suitable distance—the Wasserstein distance— the common procedure for learning the Gaussian Process model can fail due to numerical issues, arising earlier and more frequently than in the case of an Euclidean input space and, as demonstrated, impossible to avoid by adding artificial noise (nugget effect) as usually done. This paper uncovers the main reason of these issues, that is a non-stationarity relation between the Wasserstein-based squared exponential kernel and its Euclidean counterpart. As a relevant result, we learn a Gaussian Process model by assuming the input space as Euclidean and then use an algebraic transformation, based on the uncovered relation, to transform it into a non-stationary and Wasserstein-based Gaussian Process model over probability measures. This algebraic transformation is simpler than log-exp maps used on data belonging to Riemannian manifolds and recently extended to consider the pseudo-Riemannian structure of an input space equipped with the Wasserstein distance.