用双层伊辛模型实现机器学习中的结构化推断。
Ising Models with Hidden Markov Structure: Applications to Probabilistic Inference in Machine Learning
- 构建双层伊辛模型，将隐藏与观测自旋耦合建模为马尔可夫场。
- 在特定参数下存在最多三种不同平衡态，支持多解推理。
- 适用于去噪、弱监督学习等任务，适合处理层次化数据。
本文研究定义在凯莱树上的树状索引马尔可夫链（吉布斯测度），其哈密顿量耦合了两层伊辛模型：隐藏自旋 $s(x) \in \{\pm 1\}$ 和观测自旋 $σ(x) \in \{\pm 1\}$。哈密顿量包含每层内的伊辛相互作用及层间逐点发射耦合，将隐马尔可夫模型扩展为双层马尔可夫随机场。特别地，我们探讨该哈密顿量在凯莱树上的平移不变吉布斯测度（TIGM）。在模型参数满足某些显式条件时，证明最多存在三种不同的TIGM。每一类测度代表自旋系统的平衡态，为机器学习中层次化数据的推断提供了结构化方法。该模型在去噪、弱监督学习和异常检测等任务中有实际应用。凯莱树结构因其可解性，有利于精确推断。
原文摘要 · Abstract (English)
In this paper, we investigate tree-indexed Markov chains (Gibbs measures) defined by a Hamiltonian that couples two Ising layers: hidden spins \(s(x) \in \{\pm 1\}\) and observed spins \(σ(x) \in \{\pm 1\}\) on a Cayley tree. The Hamiltonian incorporates Ising interactions within each layer and site-wise emission couplings between layers, extending hidden Markov models to a bilayer Markov random field. Specifically, we explore translation-invariant Gibbs measures (TIGM) of this Hamiltonian on Cayley trees. Under certain explicit conditions on the model's parameters, we demonstrate that there can be up to three distinct TIGMs. Each of these measures represents an equilibrium state of the spin system. These measures provide a structured approach to inference on hierarchical data in machine learning. They have practical applications in tasks such as denoising, weakly supervised learning, and anomaly detection. The Cayley tree structure is particularly advantageous for exact inference due to its tractability.
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