逼近对称正定矩阵对数行列式的随机算法
A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix
摘要：
本文提出了一种新的对称正定（SPD）矩阵行列式对数的近似算法。该算法是随机化的，并利用Avron和Toledo~ \cite{AT11}的方法逼近一个特殊构造的矩阵的少量矩阵幂的迹。从理论的角度来看，我们提出了我们的算法的添加剂和相对误差界。我们的加性误差界适用于任何SPD矩阵，而我们的相对误差界适用于特征值位于区间$(θ_1,1)$中的SPD矩阵，其中$0<θ_1<1$ ;后一种设置在~ \cite{icml2015_hana15}中提出。从实证的角度来看，我们证明了我们的算法的C++实现可以近似的对数的行列式大矩阵非常准确地在几秒钟内。
英文摘要：
We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo~\cite{AT11}. From a theoretical perspective, we present additive and relative error bounds for our algorithm. Our additive error bound works for any SPD matrix, whereas our relative error bound works for SPD matrices whose eigenvalues lie in the interval $(θ_1,1)$, with $0<θ_1<1$; the latter setting was proposed in~\cite{icml2015_hana15}. From an empirical perspective, we demonstrate that a C++ implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds.
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