  Estimating the probability of rare failure events is an essential step in the
reliability assessment of engineering systems. Computing this failure
probability for complex non-linear systems is challenging, and has recently
spurred the development of active-learning reliability methods. These methods
approximate the limit-state function (LSF) using surrogate models trained with
a sequentially enriched set of model evaluations. A recently proposed method
called stochastic spectral embedding (SSE) aims to improve the local
approximation accuracy of global, spectral surrogate modelling techniques by
sequentially embedding local residual expansions in subdomains of the input
space. In this work we apply SSE to the LSF, giving rise to a stochastic
spectral embedding-based reliability (SSER) method. The resulting partition of
the input space decomposes the failure probability into a set of
easy-to-compute \rev{conditional} failure probabilities. We propose a set of
modifications that tailor the algorithm to efficiently solve rare event
estimation problems. These modifications include specialized refinement domain
selection, partitioning and enrichment strategies. We showcase the algorithm
performance on four benchmark problems of various dimensionality and complexity
in the LSF.
