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.

翻译：集成卡尔曼反演是一种基于集成的反问题求解方法。其无梯度公式使其成为处理复杂公式化问题的有吸引力的工具。然而，EKI存在"子空间特性"问题，即EKI解被限制在初始集成张成的子空间内。这意味着集成规模必须大于问题维度才能确保EKI收敛到正确解。这种集成规模缩放在实际中不可行，阻碍了EKI在高维问题中的应用。为解决该问题，我们提出一种利用dropout正则化缓解子空间问题的新方法。我们证明dropout-EKI在小集成设置下能够收敛，且算法的计算代价随维度线性增长。我们还表明dropout-EKI达到最优的查询复杂度（仅差常数因子）。数值算例验证了方法的有效性。