基于确定性三角特征的可扩展高斯过程回归：安全模型预测控制的均匀边界
Scalable Gaussian Process Regression via Deterministic Trigonometric Features: Uniform Bounds for Safe Model Predictive Control
AI总结：
本文提出确定性三角特征高斯过程（DTF-GP）框架，将GP回归转化为贝叶斯线性回归，推导其高概率一致不确定性边界，集成至学习型MPC后，在大数据场景下兼顾安全保证与计算效率。
AI中文摘要：
在存在模型失配的情况下，基于学习的模型预测控制（MPC）结合高斯过程（GP）是实现安全控制的有效方法。高概率安全保证通常要求不确定性边界在整个状态-输入域上一致成立，但现有边界仅适用于完整GP回归。由于精确GP推断的计算量随数据点数量增长而急剧增加，在大数据场景下部署并不现实。为解决这一问题，本文提出了一种可扩展的GP框架，该框架可推导得到一致不确定性边界。本文定义了确定性三角特征高斯过程（DTF-GP），这是一种基于离散化三角特征的有限维核近似方法，可将GP回归转化为特征空间中的贝叶斯线性回归。本文推导了所提出DTF-GP的高概率一致不确定性边界，并针对平方指数核情况给出了其闭式解。最后，将DTF-GP集成到基于学习的MPC方案中，结果表明，该方案在大数据场景下可提供与完整GP相当的高概率安全保证和探索性能，同时提升了计算效率。
英文摘要：
Learning-based Model Predictive Control (MPC) using Gaussian processes (GPs) is an effective approach for safe control in the presence of model mismatch. High-probability safety guarantees typically require uncertainty bounds that hold uniformly over the entire state--input domain, but existing bounds are available only for full GP regression. Since exact GP inference scales poorly with the number of data points, its deployment is impractical in large-data regimes. We close this gap by developing a scalable GP framework that admits the derivation of uniform uncertainty bounds. We formalize a deterministic trigonometric feature Gaussian process (DTF-GP), a finite-dimensional kernel approximation based on discretized trigonometric features that reduces GP regression to Bayesian linear regression in feature space. We derive a high-probability uniform uncertainty bound for the proposed DTF-GP and provide its closed-form solution for the squared-exponential kernel case. Finally, we integrate the DTF-GP into a learning-based MPC scheme and demonstrate that it provides high-probability safety guarantees and exploration performance comparable to a full GP while improving computational efficiency in large-data regimes.
