基于基张量网络中参数化张量分解的概率推断
On Probabilistic Inference Through Parametric Tensor Decomposition in Base Tensor Networks
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- University of Münster（明斯特大学）
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中文总结 AI 辅助
针对高树宽图模型中概率推断困难的问题，提出通过基张量网络表示并利用参数化张量分解控制推断可处理性的框架，刻画了可高效收缩的基张量类别。
中文摘要 AI 辅助
概率推断通常仅在低树宽图模型中易于处理，这限制了其在高树宽场景中的有效适用性。许多现有方法通过利用特定的参数化结构（如对称性）来提高效率。然而，这些方法通常要求此类结构显式存在，从而限制了它们对更广泛图模型的适用性。为解决这一局限，我们提出一个框架，其中可处理推断由潜在参数化结构利用所控制，而非要求其先验地显式存在。我们的方法首先将图模型重新参数化为一种特定的张量网络表示，我们称之为基张量网络。该表示产生两个关键性质，使得推断可处理性能够由参数化结构控制：1) 首先，推断的复杂性主要由单个张量（称为基张量）的参数化结构决定。我们刻画了几类可处理的基张量，对于这些基张量，整个基张量网络可以被高效收缩。2) 其次，分解基张量再次产生一组基张量网络。这使得推断自然地归结为将基张量分解为具有足够参数化结构的可处理组件。我们将此过程称为参数化张量分解。通过利用基张量内的参数化结构，我们的框架提供了一种超越结构显式存在场景的推断新视角。
英文摘要
Probabilistic inference is generally only tractable in low-treewidth graphical models, limiting its effective applicability in high-treewidth settings. Many existing methods improve efficiency by exploiting specific parametric structure, such as symmetries. However, they typically require such structure to be explicitly present, limiting their applicability to a broader range of graphical models. To address this limitation, we propose a framework where tractable inference is controlled by latent parametric structure exploitation, rather than requiring it to be explicitly present a priori. Our approach first reparameterises a graphical model as a specific tensor network representation, which we call a base tensor network. This representation yields two key properties that allow inference tractability to be controlled by parametric structure: 1) First, the complexity of inference is mainly determined by the parametric structure of a single tensor, called the base tensor. We characterise several tractable classes of base tensors for which the entire base tensor network can be contracted efficiently. 2) Second, decomposing the base tensor yields again a collection of base tensor networks. This allows inference to be naturally reduced to decomposing the base tensor into tractable components with sufficient parametric structure. We call this procedure parametric tensor decomposition. By exploiting parametric structure within the base tensor, our framework enables a novel view on inference beyond settings where such structure is explicitly present.
