论文标题
高斯流程对离散概率度量的回归：在欧几里得和瓦斯恒星平方之间的非平稳性关系上
Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernels
论文作者
论文摘要
高斯过程回归是在许多现实生活应用中成功采用的一种内核方法。最近，将这种方法扩展到非欧几里得输入空间的兴趣越来越大，例如本文所考虑的，包括概率指标。尽管可以通过使用合适的距离（Wasserstein距离）来定义积极的确定内核，但学习高斯过程模型的常见程序可能会因数值问题而失败，而与Euclidean Input Space更早，更频繁地引起的频率，并且在本文中，该论文在本文中无法通过添加人造噪声来避免（通常可以避免使用人工噪声）。本文揭示了这些问题的主要原因，这是基于Wasserstein的平方指数内核与其基于欧几里得的对应物之间的非平稳性关系。作为一个相关结果，通过假设输入空间为欧几里得人士，然后基于未覆盖的关系来学习高斯过程模型，然后将其转化为非平稳的基于损失的高斯流程模型，而不是概率度量。该代数转换比属于riemannian歧管的数据中使用的对数增强图更简单，并且最近扩展了以考虑配备了Wasserstein距离的输入空间的伪里曼尼亚结构。
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 in this paper, that cannot be avoided by adding artificial noise (nugget effect) as usually done. This paper uncovers the main reason of these issues, that is a non-stationarity relationship between the Wasserstein-based squared exponential kernel and its Euclidean-based counterpart. As a relevant result, the Gaussian Process model is learned by assuming the input space as Euclidean and then an algebraic transformation, based on the uncovered relation, is used 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 in the case of data belonging to Riemannian manifolds and recently extended to consider the pseudo-Riemannian structure of an input space equipped with the Wasserstein distance.
