AI 中文总结
通过最优传输将Witsenhausen反例转化为变分问题，平衡传输代价与估计误差，推导最优解性质及高斯基准，并求解MMSE正则化量化问题，揭示控制代价高低下的极限控制器形式。
AI 中文摘要
我们通过最优传输研究标量Witsenhausen反例。将第一个控制吸收进一个传输映射中，将该问题重新表述为关于目标分布$Q$的变分问题$J^=\inf_Q{k^2W_2^2(P,Q)+\operatorname{mmse}(Q)}$。该表述平衡了二次Wasserstein传输代价与最小均方估计误差，最优的第一个控制器恢复为将先验分布$P$推送到最小化分布$Q^$的单调重排。我们建立了$Q^$的存在性和绝对连续性，推导了Euler-Lagrange条件，给出了估计代价的等价Fisher信息表示，并获得了具有线性最优性显式阈值的半闭式高斯基准。将目标分布限制为有限支撑，得到一个MMSE正则化的最优量化问题。其平稳性条件将质心水平与Voronoi决策单元耦合，并在控制代价高昂时简化为经典的Lloyd-Max条件。我们使用控制惩罚$k$中的确定性退火同伦求解该非凸问题。最后，我们推导了$J^$的小$k$和大$k$渐近行为，并识别出极限控制器：当控制代价高昂时为线性控制器，当控制代价低廉时为两级信号量化器。数值结果展示了这两种机制。
英文摘要
We study the scalar Witsenhausen counterexample through optimal transport. Absorbing the first control into a transport map recasts the problem as the variational problem $J^=\inf_Q{k^2W_2^2(P,Q)+\operatorname{mmse}(Q)}$ over target laws $Q$. This formulation balances quadratic Wasserstein transport cost against minimum mean-square estimation error, with the optimal first controller recovered as the monotone rearrangement that pushes the prior $P$ to a minimizing law $Q^$. We establish existence and absolute continuity of $Q^$, derive an Euler-Lagrange condition, give an equivalent Fisher-information representation of the estimation cost, and obtain a semi-closed-form Gaussian benchmark with an explicit threshold for linear optimality. Restricting the target law to finite support yields an MMSE-regularized optimal quantization problem. Its stationarity conditions couple centroid levels with Voronoi decision cells and reduce to the classical Lloyd-Max conditions when control is expensive. We solve this nonconvex problem using a deterministic-annealing homotopy in the control penalty $k$. Finally, we derive the small- and large-$k$ asymptotics of $J^$ and identify the limiting controllers: a linear controller when control is expensive and a two-level signaling quantizer when it is cheap. Numerical results illustrate both regimes.
