Title: On a novel spherical Monte Carlo method via group representation
Authors: Huei-Wen Teng - National Chiao Tung University (Taiwan) [presenting]
Ming-Hsuan Kang - National Chiao Tung University (Taiwan)
Runze Li - The Pennsylvania State University (United States)
Abstract: Accurate and efficient calculation of $d$-dimensional integrals for large $d$ is of crucial importance in various scientific disciplines. Via spherical transformation, standard spherical Monte Carlo estimators consist of independent radii and a set of unit vectors uniformly distributed on a unit sphere. A random orthogonal group is used to rotate a set of unit vectors simultaneously, and can be generated by applying the Gram-Schmit procedure to a $d\times d$ matrix with i.i.d. standard normal random variables as entries. The generation of a random orthogonal group is however computationally demanding. To overcome this problem, a novel spherical Monte Carlo approach is proposed via group representation: By constructing a subgroup of the orthogonal groups, the spherical integral is calculated using the group orbit of a random unit vector. In this case, the generation of a random unit vector only needs $d$ i.i.d. standard normal random variable. The proposed method outperforms existing methods in terms of computation efficiency in high-dimensional cases. Theoretical properties of the proposed subset are provided. Extensive numerical experiments with applications in finance confirm our claims.