提出无需显式平方即可高效计算电路边缘分布的新方法
How to Square Tensor Networks and Circuits Without Squaring Them
- 用酉矩阵参数化平方电路，避免复杂计算
- 在不损失表达能力的前提下实现快速边缘化
- 适合需要高效概率推断的机器学习任务
平方张量网络（TNs）及其扩展——平方电路——被用作表达性强的概率分布估计器，支持闭式边缘化。然而，平方操作会增加计算配分函数或边缘化变量的复杂度，限制了其在机器学习中的应用。现有方法通过酉矩阵参数化张量网络的规范形式以简化边缘化计算，但该方法不适用于电路，因为电路可表示无法直接映射到已知张量网络的因式分解。受规范形式中正交性与电路中确定性支持可处理极大化的启发，本文提出对平方电路进行参数化，以克服边缘化开销。该参数化使即使在非张量网络结构的因式分解中也能实现高效边缘化，而这些结构原本边缘化计算困难。实验表明，所提条件在分布估计任务中无表达力损失，同时提升学习效率。
原文摘要 · Abstract (English)
Squared tensor networks (TNs) and their extension as computational graphs--squared circuits--have been used as expressive distribution estimators, yet supporting closed-form marginalization. However, the squaring operation introduces additional complexity when computing the partition function or marginalizing variables, which hinders their applicability in ML. To solve this issue, canonical forms of TNs are parameterized via unitary matrices to simplify the computation of marginals. However, these canonical forms do not apply to circuits, as they can represent factorizations that do not directly map to a known TN. Inspired by the ideas of orthogonality in canonical forms and determinism in circuits enabling tractable maximization, we show how to parameterize squared circuits to overcome their marginalization overhead. Our parameterizations unlock efficient marginalization even in factorizations different from TNs, but encoded as circuits, whose structure would otherwise make marginalization computationally hard. Finally, our experiments on distribution estimation show how our proposed conditions in squared circuits come with no expressiveness loss, while enabling more efficient learning.
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