Gaussian process regression in the flat limit
平坦极限下的高斯过程回归AI 解读 ↓
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平坦极限下的高斯过程回归
- 研究问题
- 论文研究固定样本下核长度尺度趋于无穷时，高斯过程回归的预测均值、方差及后验预测分布会趋向何种模型。
- 方法与思路
- 作者利用核矩阵在平坦极限下的谱性质，引入半参数模型的预测等价概念，统一分析不同核对应的回归极限。
- 材料陈述的贡献
- 摘要称预测均值和方差可趋于多项式或多调和样条回归，并指出留一损失最优预测可能位于数值实现难以探索的大尺度处。
- 阅读关注点
- 应核对核类型、极限过程及预测等价的定义，确认相关模型参数的处理方式，并检查大长度尺度下的数值困难。
- 阅读边界
- 索引摘要版本未注明；具体核条件和极限设定未说明，固定样本结论不能替代大样本保证，也不意味着所有核具有相同极限。
原文摘要
Gaussian process (GP) regression is a fundamental tool in Bayesian statistics. It is also known as kriging and is the Bayesian counterpart to the frequentist kernel ridge regression. Most of the theoretical work on GP regression has focused on a large-n asymptotics, characterising the behaviour of GP regression as the amount of data increases. Fixed-sample analysis is much more difficult outside of simple cases, such as locations on a regular grid. In this work, we perform a fixed-sample analysis that was first studied in the context of approximation theory by Fornberg and Driscoll (2002), called the “flat limit”. In flat-limit asymptotics, the goal is to characterise kernel methods as the length-scale of the kernel function tends to infinity, so that kernels appear flat over the range of the data. Surprisingly, this limit is well-defined, and displays interesting behaviour: Driscoll and Fornberg showed that radial basis interpolation converges in the flat limit to polynomial interpolation, if the kernel is Gaussian. Subsequent work showed that this holds true in the multivariate setting as well, but that kernels other than the Gaussian may have (polyharmonic) splines as the limit interpolant. Leveraging recent results on the spectral behaviour of kernel matrices in the flat limit, we study the flat limit of Gaussian process regression. Results show that Gaussian process regression tends in the flat limit to (multivariate) polynomial regression, or (polyharmonic) spline regression, depending on the kernel. Importantly, this holds for both the predictive mean and the predictive variance, so that the posterior predictive distributions become equivalent. For the proof, we introduce the notion of prediction-equivalence of semiparametric models, which lets us state flat-limit results in a compact and unified manner. Our results have practical consequences: for instance, they show that optimal GP predictions in the sense of leave-one-out loss may occur at very large length-scales, which would be invisible to current implementations because of numerical difficulties.