2021/07/13 by Maxime El-Masri, Jérôme Morio, ElMasri, Maxime +3
Mathematics · Engineering · Decision Sciences · #Random Matrices and Applications #Geophysical Methods and Applications #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2107.06091
In this paper we propose a dimension-reduction strategy in order to improve\nthe performance of importance sampling in high dimension. The idea is to\nestimate variance terms in a small number of suitably chosen directions. We\nfirst prove that the optimal directions, i.e., the ones that minimize the\nKullback--Leibler divergence with the optimal auxiliary density, are the\neigenvectors associated to extreme (small or large) eigenvalues of the optimal\ncovariance matrix. We then perform extensive numerical experiments that show\nthat as dimension increases, these directions give estimations which are very\nclose to optimal. Moreover, we show that the estimation remains accurate even\nwhen a simple empirical estimator of the covariance matrix is used to estimate\nthese directions. These theoretical and numerical results open the way for\ndifferent generalizations, in particular the incorporation of such ideas in\nadaptive importance sampling schemes.\n