Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding
Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding
复制标题
通过最优传输和希尔伯特嵌入具有多维分布输入的高斯过程
DOI：
10.13140/rg.2.2.23440.92165
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发表时间：
2018
影响因子：
1.1
通讯作者：
V. Spokoiny
中科院分区：
文献类型：
--
作者：
D. Ginsbourger;F. Bachoc;A. Suvorikova;Jean;V. Spokoiny
在这项工作中，我们研究由多维分布索引的高斯过程。虽然基于 Wasserstein 距离直接构造径向正定核已被证明在一维情况下是可能的，但这种构造不能很好地扩展到我们在此说明的多维情况。为了解决基于最优传输在多元分布之间定义正定核的问题，我们转而求助于依赖于参考分布的最优传输映射的希尔伯特空间嵌入，我们建议将其作为 Wasserstein 重心。我们依次表征希尔伯特空间上的径向正定核，并表明几乎所有协方差函数参数族的协方差参数在（无限维）希尔伯特空间的情况下都是微遍历的。我们还研究了我们建议的正定核在多维分布上的统计特性，重点是当总体 Wasserstein 重心被经验重心替换时的一致性以及高斯分布特殊情况下的附加显式结果。最后，我们基于我们建议的多维分布输入回归问题中的正定核、基于来自综合示例和机械工程测试用例的模拟数据来研究高斯过程方法。
In this work, we investigate Gaussian Processes indexed by multidimensional distributions. While directly constructing radial positive definite kernels based on the Wasserstein distance has been proven to be possible in the unidimensional case, such constructions do not extend well to the multidimensional case as we illustrate here. To tackle the problem of defining positive definite kernels between multivariate distributions based on optimal transport, we appeal instead to Hilbert space embeddings relying on optimal transport maps to a reference distribution, that we suggest to take as a Wasserstein barycenter. We characterize in turn radial positive definite kernels on Hilbert spaces, and show that the covariance parameters of virtually all parametric families of covariance functions are microergodic in the case of (infinite-dimensional) Hilbert spaces. We also investigate statistical properties of our suggested positive definite kernels on multidimensional distributions, with a focus on consistency when a population Wasserstein barycenter is replaced by an empirical barycenter and additional explicit results in the special case of Gaussian distributions. Finally, we study the Gaussian process methodology based on our suggested positive definite kernels in regression problems with multidimensional distribution inputs, on simulation data stemming both from synthetic examples and from a mechanical engineering test case.
影响因子：
4.8
作者：
Lajoie, Bryan R.;Dekker, Job;Kaplan, Noam
通讯作者：
Kaplan, Noam