A randomized algorithm for approximating the log determinant of a symmetric positive definite matrix

作者：Boutsidis Christos; Drineas Petros^*; Kambadur Prabhanjan; Kontopoulou Eugenia Maria^*; Zouzias Anastasios

来源：Linear Algebra and Its Applications, 2017, 533: 95-117.

DOI：10.1016/j.laa.2017.07.004

• Logdeterminant approximation
• SPD matrix
• Randomized algorithm
• Power method
• Trace estimation

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 [1]. 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 (theta(1),1), with 0 < theta(1) < 1; the latter setting was proposed in [16]. 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.

• 出版日期2017-11-15