AI 中文总结
针对双域线性因果信号，提出一种新的DAG乘积算子，使乘积图上的SEM、傅里叶模式、因果移位及滤波器均可按因子分解，避免求逆大型传递闭包矩阵。
AI 中文摘要
我们为以两个有向无环图（DAG）的乘积为索引、并由线性结构方程模型（SEM）描述的信号开发了一套信号处理框架。这种设置出现在（线性）因果关系沿两个域作用的情形中，例如组件与制造阶段、或基因与实验条件。若忽略底层图的分解和固有的双轴因果结构，则傅里叶分析需要求逆一个加权传递闭包矩阵，其大小为两个因子大小的乘积。鉴于标准图乘积无法产生可分解的传递闭包，我们引入了一种新的DAG乘积，在该乘积下可分离性成立。我们在顶点域中阐述这一新算子，并证明它同时使乘积DAG上的SEM、傅里叶模式、因果移位和滤波器在其组成图因子上可分离。
英文摘要
We develop a signal processing framework for signals indexed by the product of two directed acyclic graphs (DAGs) and described by a linear structural equation model (SEM). Such a setup arises whenever (linear) causal relations act along two domains, as in component versus manufacturing stage or gene versus experimental condition. Disregarding the factorization of the underlying graph and the native two-axis causal structure requires inverting a weighted transitive closure matrix whose size is the product of the two factor sizes for Fourier analysis. Recognizing that standard graph products fail to yield factorizable transitive closures, we introduce a new DAG product under which separability holds. We motivate the new operator in the vertex domain, and show that it also renders the SEM, the Fourier modes, the causal shifts, and the filters on the product DAG separable across its constituent graph factors.
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