论文标题
插值log-determinant和Matrix $ \ mathbf {a} + t \ mathbf {b} $的幂的痕迹
Interpolating Log-Determinant and Trace of the Powers of Matrix $\mathbf{A} + t \mathbf{B}$
论文作者
论文摘要
我们为功能开发启发式插插方法$ t \ mapsto \ log \ det \ left（\ mathbf {a} + t \ t \ mathbf {b} \ right）$和$ t \ t \ mapsto \ mapsto \ operatatOrname {trace}矩阵$ \ mathbf {a} $和$ \ mathbf {b} $是Hermitian和正（半）确定的，$ p $和$ t $是实际变量。这些功能在统计，机器学习和计算物理学的许多应用中都有特征。提出的插值函数基于对这些功能的锐度界限的修改。我们通过数值示例证明了所提出的方法的准确性和性能，即高斯过程回归的边际最大似然估计以及用广义交叉验证方法对岭回归的正则参数的估计。
We develop heuristic interpolation methods for the functions $t \mapsto \log \det \left( \mathbf{A} + t \mathbf{B} \right)$ and $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{p} \right)$ where the matrices $\mathbf{A}$ and $\mathbf{B}$ are Hermitian and positive (semi) definite and $p$ and $t$ are real variables. These functions are featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of sharp bounds for these functions. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.
