考虑输入测量误差的高斯过程回归，提升不确定性量化可靠性。
Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty
- 用沃尔德斯坦距离建模输入不确定性，将噪声输入视为概率分布
- 提出PWA核函数，支持高效计算且保证正定性，避免引入隐变量
- 适用于需要透明、鲁棒不确定性的工业建模与决策场景
高斯过程回归广泛用于不确定性量化，但标准形式假设输入无误差。当输入存在测量误差时，这种误变量（EIV）设置会导致后验区间过窄和决策偏差。本文通过将每个噪声输入表示为概率测度，并利用测度间的沃尔德斯坦距离定义协方差，研究了输入不确定性下的高斯过程回归。基于此视角，我们提出一种确定性投影沃尔德斯坦自适应尺度（PWA）核，其一维分量具有闭式表达，乘积结构保证了在分布上的可扩展性和正定性。与潜在输入高斯过程模型不同，基于PWA的高斯过程（PWAGPs）无需引入未观测协变量或蒙特卡洛投影，使不确定性量化更透明、更稳健。
原文摘要 · Abstract (English)
Gaussian process (GP) regression is widely used for uncertainty quantification, yet the standard formulation assumes noise-free covariates. When inputs are measured with error, this errors-in-variables (EIV) setting can lead to optimistically narrow posterior intervals and biased decisions. We study GP regression under input measurement uncertainty by representing each noisy input as a probability measure and defining covariance through Wasserstein distances between these measures. Building on this perspective, we instantiate a deterministic projected Wasserstein ARD (PWA) kernel whose one-dimensional components admit closed-form expressions and whose product structure yields a scalable, positive-definite kernel on distributions. Unlike latent-input GP models, PWA-based GPs (\PWAGPs) handle input noise without introducing unobserved covariates or Monte Carlo projections, making uncertainty quantification more transparent and robust.
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