Improvement of the cross-entropy method in high dimension through a one-dimensional projection without gradient estimation

Rare event probability estimation is an important topic in reliability analysis. Stochastic methods, such as importance sampling, have been developed to estimate such probabilities but they often fail in high dimension. In this paper, we propose a simple cross-entropy-based importance sampling algorithm to improve rare event estimation in high dimension. We consider the cross-entropy method with Gaussian auxiliary distributions and we suggest to update the Gaussian covariance matrix only in a one-dimensional subspace. The idea is to project in the one dimensional subspace spanned by the Gaussian mean, which gives an influential direction for the variance estimation. This approach does not require any additional simulation budget compared to the basic cross-entropy algorithm and greatly improves its performance in high dimension, as we show on different numerical test cases.