We investigate the use of Wasserstein gradient flows for finding an $E$-optimal design for a regression model. Unlike the commonly used $D$- and $L$-optimality criteria, the $E$-criterion finds a design that maximizes the smallest eigenvalue of the information matrix, and so it is a non-differentiable criterion unless the minimum eigenvalue has geometric multiplicity equals to one. Such maximin design problems abound in statistical applications and present unique theoretical and computational challenges. Building on the differential structure of the $2$-Wasserstein space, we derive explicit formulas for the Wasserstein gradient of the $E$-optimality criterion in the simple-eigenvalue case. For higher multiplicities, we propose a Wasserstein steepest ascent direction and show that it can be computed exactly via a semidefinite programming (SDP) relaxation. We develop particle approximations that connect infinite-dimensional flows with finite-dimensional optimization, and provide approximation guarantees for empirical measures. Our framework extends naturally to constrained designs via projected Wasserstein gradient flows. Numerical experiments demonstrate that the proposed methods successfully recover $E$-optimal designs for both linear and nonlinear regression models, with competitive accuracy and scalability compared to existing heuristic approaches. This work highlights the potential of optimal transport-based dynamics as a unifying tool for studying challenging optimal design problems.

翻译：本文研究了利用Wasserstein梯度流为回归模型寻找E最优设计的方法。与常用的D最优和L最优准则不同，E准则旨在最大化信息矩阵最小特征值，因此除非最小特征值的几何重数为一，否则该准则不可微。此类极大极小设计问题在统计应用中广泛存在，并带来独特的理论与计算挑战。基于2-Wasserstein空间的微分结构，我们在单特征值情形下推导了E最优准则的Wasserstein梯度显式公式。对于更高重数情形，我们提出了一种Wasserstein最速上升方向，并证明可通过半定规划松弛精确计算。我们建立了连接无限维流与有限维优化的粒子近似方法，并为经验测度提供了近似保证。通过投影Wasserstein梯度流，我们的框架可自然扩展至约束设计问题。数值实验表明，所提方法在线性与非线性回归模型中均能成功恢复E最优设计，与现有启发式方法相比具有竞争力的精度与可扩展性。本工作凸显了基于最优传输的动态学作为研究挑战性最优设计问题的统一工具的潜力。