  Ensemble Kalman inversion (EKI) is an ensemble-based method to solve inverse
problems. Its gradient-free formulation makes it an attractive tool for
problems with involved formulation. However, EKI suffers from the ''subspace
property'', i.e., the EKI solutions are confined in the subspace spanned by the
initial ensemble. It implies that the ensemble size should be larger than the
problem dimension to ensure EKI's convergence to the correct solution. Such
scaling of ensemble size is impractical and prevents the use of EKI in high
dimensional problems. To address this issue, we propose a novel approach using
dropout regularization to mitigate the subspace problem. We prove that
dropout-EKI converges in the small ensemble settings, and the computational
cost of the algorithm scales linearly with dimension. We also show that
dropout-EKI reaches the optimal query complexity, up to a constant factor.
Numerical examples demonstrate the effectiveness of our approach.
