Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.
翻译:高斯进程是机器学习模型,能够以具有不确定性的方式学习未知功能,从而便利建立最佳决策系统。由于希望在新的科学领域部署高斯进程,迅速增长的研究线侧重于建设性地扩大这些模型,以处理非欧洲的域,包括里曼尼主义的元体,如球体和托里等。我们建议采用一些技术,在里曼尼主义多元体上将这一类技术推广到向量场的模型,这对物理科学的一些应用领域很重要。为了做到这一点,我们提出了一个用于建造测量独立内核的一般配方,以引导高斯向量体的向量场,即矢量价值高斯进程与地理测量相一致,从斯卡拉价值的里曼内核内核。我们将标准高斯过程培训方法,例如变异推断方法,推广到这一环境。这样,里曼尼多元体的矢量值高斯进程就能够使用标准方法进行培训,并使机器学习实践者能够利用这些方法。