Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for broadening the applicability of Gaussian processes. This is done in two ways. Firstly, we develop pathwise conditioning techniques for Gaussian processes, which allow one to express posterior random functions as prior random functions plus a dependent update term. We introduce a wide class of efficient approximations built from this viewpoint, which can be randomly sampled once in advance, and evaluated at arbitrary locations without any subsequent stochasticity. This key property improves efficiency and makes it simpler to deploy Gaussian process models in decision-making settings. Secondly, we develop a collection of Gaussian process models over non-Euclidean spaces, including Riemannian manifolds and graphs. We derive fully constructive expressions for the covariance kernels of scalar-valued Gaussian processes on Riemannian manifolds and graphs. Building on these ideas, we describe a formalism for defining vector-valued Gaussian processes on Riemannian manifolds. The introduced techniques allow all of these models to be trained using standard computational methods. In total, these contributions make Gaussian processes easier to work with and allow them to be used within a wider class of domains in an effective and principled manner. This, in turn, makes it possible to potentially apply Gaussian processes to novel decision-making settings.
翻译:使用 Gaussian 进程进行的巴伊西亚学习为决策提供了一个基础框架, 以平衡通过收集数据可以学到的东西来平衡已知的东西。 在这项论文中, 我们开发了扩大高萨进程应用性的技术。 这是以两种方式完成的。 首先, 我们开发了高萨进程路径化调节技术, 使一个人能够以先前随机函数和一个依赖性更新术语来表达事后随机函数。 我们引入了从这个角度构建的范围广泛的高效近似框架, 可以事先随机抽样一次, 并在任意地点进行评估, 而不产生任何后续的随机性 。 关键属性提高了效率, 使得在决策设置中部署高萨进程模型更加简单。 其次, 我们开发了高萨进程模型, 包括Riemannian 的矩阵和图表。 我们从一个完全具有建设性的表达方式, 在 Riemann 的方位和图表上随机随机抽样地标, 在这些概念上, 我们用经过培训的多层次模型 来定义高萨进程, 并用这些高萨进程在全部的矢量模型中 。