We introduce Deep Jump Gaussian Processes (DJGP), a novel method for surrogate modeling of a piecewise continuous function on a high-dimensional domain. DJGP addresses the limitations of conventional Jump Gaussian Processes (JGP) in high-dimensional input spaces by integrating region-specific, locally linear projections with JGP modeling. These projections employ region-dependent matrices to capture local low-dimensional subspace structures, making them well suited to the inherently localized modeling behavior of JGPs, a variant of local Gaussian processes. To control model complexity, we place a Gaussian Process prior on the projection matrices, allowing them to evolve smoothly across the input space. The projected inputs are then modeled with a JGP to capture piecewise continuous relationships with the response. This yields a distinctive two-layer deep learning of GP/JGP. We further develop a scalable variational inference algorithm to jointly learn the projection matrices and JGP hyperparameters. Rigorous theoretical analysis and extensive empirical studies are provided to justify the proposed approach. In particular, we derive an oracle error bound for DJGP and decompose it into four distinct sources of error, which are then linked to practical implications. Experiments on synthetic and benchmark datasets demonstrate that DJGP achieves superior predictive accuracy and more reliable uncertainty quantification compared with existing methods.

翻译：本文提出深度跳跃高斯过程（DJGP），一种用于高维域上分段连续函数代理建模的新方法。DJGP通过将区域特定的局部线性投影与JGP建模相结合，解决了传统跳跃高斯过程（JGP）在高维输入空间中的局限性。这些投影采用区域依赖矩阵来捕捉局部低维子空间结构，使其非常适合JGPs（局部高斯过程的一种变体）固有的局部建模特性。为控制模型复杂度，我们在投影矩阵上设置高斯过程先验，使其能在输入空间中平滑演化。随后使用JGP对投影后的输入进行建模，以捕捉其与响应变量的分段连续关系。这形成了一种独特的GP/JGP双层深度学习架构。我们进一步开发了可扩展的变分推断算法，以联合学习投影矩阵和JGP超参数。本文提供了严格的理论分析和广泛的实证研究以验证所提方法的有效性。特别地，我们推导了DJGP的预言误差界，并将其分解为四个独立的误差来源，进而与实际应用意义建立关联。在合成数据集和基准数据集上的实验表明，相较于现有方法，DJGP在预测精度和不确定性量化可靠性方面均表现出优越性能。