Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization

The main motivation behind ensemble Kalman methods is that they often perform well with a small ensemble size N, which is essential in applications where generating each particle is costly. However, theoretical studies have primarily focused on large ensemble asymptotics, that is, on the limit N . While these mean-field results are mathematically interesting and have led to significant practical improvements, they fail to explain the empirical success of ensemble Kalman methods when deployed with a small ensemble size. The aim of this paper is to develop a non-asymptotic analysis of ensemble Kalman updates that rigorously explains why, and under what circumstances, a small ensemble size may suffice. To that end, we establish non-asymptotic error bounds in terms of suitable notions of effective dimension of the prior covariance model that account for spectrum decay (which may represent smoothness of a prior random field) and approximate sparsity (which may represent spatial decay of correlations).