We present a differentiable framework for end-to-end mutual information (MI) optimization over linear Gaussian directed acyclic graphs (DAGs). The framework targets network-wide design under global constraints, such as a total transmit power budget, and covers MIMO precoding, amplify-and-forward relays, RIS-aided channels, and branching/merging topologies within a common linear Gaussian model. Its core ingredient is a \emph{K-recursion} that analytically propagates all node-pair covariances along the DAG in topological order, including non-adjacent cross-covariances that are necessary for correctly handling branching and merging paths. The resulting covariances yield a closed-form log-determinant expression for the end-to-end MI as a smooth function of the controllable factors. Complex-valued reverse-mode automatic differentiation on this K-recursion then returns the exact Wirtinger gradient at every controllable factor in a single backward sweep, and projected gradient ascent (PGA) is used to maximize the MI under the global constraints. Because no closed-form gradient expression per topology is required, the same topology-agnostic implementation applies to any linear Gaussian DAG. A single topology-agnostic implementation is applied to four representative DAG classes: single-link MIMO, a diamond DAG, a two-hop AF relay, and input-covariance shaping. The same implementation reaches the classical water-filling optimum in the settings where it is available and yields MI improvements in non-single-link topologies without using topology-specific gradient formulas. A further experiment on a multi-layer Gaussian network (11 nodes, 5 layers) illustrates applicability to nontrivial multi-layer topologies for which no closed-form gradient is available.

翻译：我们提出了一个可微分框架，用于在无环有向图上实现端到端互信息优化。该框架针对全局约束（如总发射功率预算）下的网络级设计，涵盖MIMO预编码、放大转发中继、RIS辅助信道以及分支/合并拓扑等常见线性高斯模型。其核心要素是一种K-递归算法，该算法按拓扑顺序沿DAG解析传播所有节点对协方差，包括正确处理分支与合并路径所需的非相邻交叉协方差。由此产生的协方差可得到端到端互信息的闭式对数行列式表达式，该表达式是可控因子的光滑函数。在此K-递归上执行复值反向模式自动微分，即可在单次反向扫描中返回每个可控因子处的精确Wirtinger梯度，并采用投影梯度上升法在全局约束下最大化互信息。由于无需针对每种拓扑预先推导闭式梯度表达式，同一与拓扑无关的实现方法可适用于任意线性高斯DAG。我们将这种与拓扑无关的统一实现应用于四类代表性DAG：单链路MIMO、菱形DAG、两跳AF中继以及输入协方差整形。该实现可在适用场景达到经典注水最优解，并在非单链路拓扑中无需使用拓扑特定梯度公式即获得互信息提升。进一步在多层高斯网络（11节点，5层）上的实验验证了该方法在无闭式梯度可用的复杂多层拓扑中的适用性。