通过正交化提升平方电路的边缘化效率，不损失表达能力。
On Faster Marginalization with Squared Circuits via Orthonormalization
- 用正交化参数化平方电路，保证分布已归一化。
- 新算法边缘化速度优于已有方法，计算更高效。
- 适用于需要高效边缘化的高维概率建模任务。
平方张量网络（TN）及其推广形式——参数化计算图（平方电路），近年来被用于高维数据的概率分布估计。然而，平方操作在变量边缘化或计算归一化常数时引入额外复杂度，限制了其在机器学习中的应用。主流TN的规范形式通过酉矩阵参数化以简化特定边缘计算，但无法直接映射到一般电路，因后者未必对应已知的TN结构。受此启发，本文提出一种新的参数化方法，使平方电路天然编码归一化分布，并基于此设计出更高效的任意边缘计算算法。最后，理论证明该参数化对多种电路类别无表达能力损失。
原文摘要 · Abstract (English)
Squared tensor networks (TNs) and their generalization as parameterized computational graphs -- squared circuits -- have been recently used as expressive distribution estimators in high dimensions. However, the squaring operation introduces additional complexity when marginalizing variables or computing the partition function, which hinders their usage in machine learning applications. Canonical forms of popular TNs are parameterized via unitary matrices as to simplify the computation of particular marginals, but cannot be mapped to general circuits since these might not correspond to a known TN. Inspired by TN canonical forms, we show how to parameterize squared circuits to ensure they encode already normalized distributions. We then use this parameterization to devise an algorithm to compute any marginal of squared circuits that is more efficient than a previously known one. We conclude by formally showing the proposed parameterization comes with no expressiveness loss for many circuit classes.
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