We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.

中文翻译：

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内插矩阵的逆矩阵 $\mathbf{A} + t \mathbf{B}$

我们为函数 $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$ 开发了启发式插值方法，其中矩阵 $\mathbf {A}$ 和 $\mathbf{B}$ 是对称正定的，$t$ 是实变量。该函数在统计学、机器学习和计算物理学的许多应用中都有特色。所呈现的插值函数基于对我们为该函数导出的尖锐上限的修改，这是矩阵的新迹不等式。我们用数值例子证明了所提出方法的准确性和性能，即线性高斯过程回归的边际最大似然估计和使用广义交叉验证方法的岭回归正则化参数的估计。