Applications of Trace Estimation Techniques
Applications of Trace Estimation Techniques
复制标题
迹线估计技术的应用
DOI：
10.1007/978-3-319-97136-0_2
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Y. Saad
中科院分区：
文献类型：
--
作者：
Shashanka Ubaru;Y. Saad
我们讨论了迹估计技术在计算\（\mathtt {tr}（f（A））\）形式的函数中的各种应用，其中f是某个函数。我们考虑的第一个问题，可以在这种形式下铸造是近似矩阵的谱密度或态密度（DOS）。DOS是一种概率密度分布，它度量了在真实的直线上的给定点处找到矩阵的本征值的可能性，并且它是固态物理中的一个重要函数。我们还提出了一些非标准的应用程序的频谱密度。我们讨论的其他迹估计问题包括估计逆矩阵\（\mathtt {tr}（A^{-1}）\）的迹，计算特征值和估计秩的问题，以及近似对数行列式（对数函数的迹）。我们还讨论了机器学习应用中出现的一些类似计算。我们回顾了两种计算成本低的方法来计算矩阵函数的迹，即Chebyshev展开和Lanczos求积方法。给出了几个数值例子来说明这些方法在不同应用中的性能。
We discuss various applications of trace estimation techniques for evaluating functions of the form \(\mathtt {tr}(f(A))\) where f is certain function. The first problem we consider that can be cast in this form is that of approximating the Spectral density or Density of States (DOS) of a matrix. The DOS is a probability density distribution that measures the likelihood of finding eigenvalues of the matrix at a given point on the real line, and it is an important function in solid state physics. We also present a few non-standard applications of spectral densities. Other trace estimation problems we discuss include estimating the trace of a matrix inverse \(\mathtt {tr}(A^{-1})\), the problem of counting eigenvalues and estimating the rank, and approximating the log-determinant (trace of log function). We also discuss a few similar computations that arise in machine learning applications. We review two computationally inexpensive methods to compute traces of matrix functions, namely, the Chebyshev expansion and the Lanczos Quadrature methods. A few numerical examples are presented to illustrate the performances of these methods in different applications.
DOI：
10.1073/pnas.97.18.10101
发表时间：
2000-08-29
影响因子：
11.1
作者：
Alter, O;Brown, PO;Botstein, D
通讯作者：
Botstein, D