Efficient reduced-rank methods for Gaussian processes with eigenfunction expansions
Efficient reduced-rank methods for Gaussian processes with eigenfunction expansions
复制标题
具有特征函数展开的高斯过程的高效降阶方法
DOI：
10.1007/s11222-022-10124-z
复制
发表时间：
2021
影响因子：
2.2
通讯作者：
M. O’Neil
中科院分区：
文献类型：
--
作者：
P. Greengard;M. O’Neil
在这项工作中，我们介绍了高斯过程回归的降阶算法。我们的数值方案将用户指定区间上的高斯过程转换为其karhunen - lo<e:1>展开，即$$L^2$$ l2 -最优降阶表示。karhunen - lo<e:1>展开式的数值计算在预计算期间执行一次，涉及计算一个积分算子的数值特征分解，其核是高斯过程的协方差函数。karhunen - lo<s:1>展开与观测数据无关，仅取决于协方差核和高斯过程定义的区间的大小。本文的方案不要求协方差核的平移不变性。本文还介绍了一类超参数贝叶斯拟合的快速算法，并通过一维和二维的数值实验证明了算法的性能。高维的扩展在数学上是直接的，但却受到高维的标准诅咒的影响。
In this work, we introduce a reduced-rank algorithm for Gaussian process regression. Our numerical scheme converts a Gaussian process on a user-specified interval to its Karhunen–Loève expansion, the $$L^2$$ L 2 -optimal reduced-rank representation. Numerical evaluation of the Karhunen–Loève expansion is performed once during precomputation and involves computing a numerical eigendecomposition of an integral operator whose kernel is the covariance function of the Gaussian process. The Karhunen–Loève expansion is independent of observed data and depends only on the covariance kernel and the size of the interval on which the Gaussian process is defined. The scheme of this paper does not require translation invariance of the covariance kernel. We also introduce a class of fast algorithms for Bayesian fitting of hyperparameters and demonstrate the performance of our algorithms with numerical experiments in one and two dimensions. Extensions to higher dimensions are mathematically straightforward but suffer from the standard curses of high dimensions.
DOI：
10.1137/17m1161853
发表时间：
2019-01
期刊：
SIAM Rev.
影响因子：
--
作者：
Silviu-Ioan Filip;Aurya Javeed;L. Trefethen
通讯作者：
Silviu-Ioan Filip;Aurya Javeed;L. Trefethen
影响因子：
5.8
作者：
Carpenter, Bob;Gelman, Andrew;Riddell, Allen
通讯作者：
Riddell, Allen