This paper primarily considers the robust estimation problem under Wasserstein distance constraints on the parameter and noise distributions in the linear measurement model with additive noise, which can be formulated as an infinite-dimensional nonconvex minimax problem. We prove that the existence of a saddle point for this problem is equivalent to that for a finite-dimensional minimax problem, and give a counterexample demonstrating that the saddle point may not exist. Motivated by this observation, we present a verifiable necessary and sufficient condition whose parameters can be derived from a convex problem and its dual. Additionally, we also introduce a simplified sufficient condition, which intuitively indicates that when the Wasserstein radii are small enough, the saddle point always exists. In the absence of the saddle point, we solve an finite-dimensional nonconvex minimax problem, obtained by restricting the estimator to be linear. Its optimal value establishes an upper bound on the robust estimation problem, while its optimal solution yields a robust linear estimator. Numerical experiments are also provided to validate our theoretical results.

翻译：本文主要研究带加性噪声的线性测量模型中参数与噪声分布受Wasserstein距离约束的鲁棒估计问题，该问题可表述为无限维非凸极小极大问题。我们证明了该问题鞍点的存在性等价于一个有限维极小极大问题鞍点的存在性，并给出反例表明鞍点可能不存在。基于这一观察，我们提出了一个可验证的充要条件，其参数可通过凸问题及其对偶问题导出。此外，我们还引入了一个简化的充分条件，直观表明当Wasserstein半径足够小时鞍点必然存在。在鞍点不存在的情况下，我们通过将估计器限制为线性形式，求解由此得到的有限维非凸极小极大问题。其最优值为原鲁棒估计问题提供了上界，而最优解则产生一个鲁棒线性估计器。数值实验也验证了我们的理论结果。