高斯过程特征函数展开的有效降秩方法
Efficient reduced-rank methods for Gaussian processes with eigenfunction expansions
摘要：
在这项工作中，我们介绍了一个降秩算法的高斯过程回归。我们的数值方案将用户指定的区间上的高斯过程转换为其Karhunen-Loève展开，即$L^2$ -最优降秩表示。Karhunen-Loève展开的数值计算在预计算期间执行一次，并且涉及计算积分算子的数值特征分解，该积分算子的核是高斯过程的协方差函数。Karhunen-Loève展开与观测数据无关，仅取决于协方差核和定义高斯过程的区间大小。本文的方案不要求协方差核具有平移不变性。我们还介绍了一类快速算法贝叶斯拟合的超参数，并证明了我们的算法的性能与数值实验在一维和二维。向更高维度的扩展在数学上是简单的，但却遭受着高维的标准诅咒。
英文摘要：
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 $L^2$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
