用有限统计量高效学习伊辛模型参数，突破观测与计算的权衡瓶颈。
Computationally sufficient statistics for Ising models
- 通过观测不超过γ阶的统计量实现参数重建
- 对ℓ₁宽度为γ的模型，只需O(γ)阶统计量即可还原结构与耦合参数
- 当已知模型结构时，可进一步降低观测需求，适合物理系统建模
使用仅包含充分统计量的方法学习吉布斯分布长期以来被认为是计算难题。而现有的计算高效算法则依赖于从模型中生成的完整样本配置。对于许多物理系统而言，期望获得完整样本并不现实，因此需要在有限统计量下设计计算高效的求解方法。本文以伊辛模型为例，研究计算能力与观测能力之间的权衡。我们证明：对于ℓ₁宽度为γ的模型，仅需观测至O(γ)阶的统计量即可实现参数重构，从而推断模型结构、耦合系数及磁场。此外，当模型结构已知时，可在更弱的观测条件下高效完成学习任务。
原文摘要 · Abstract (English)
Learning Gibbs distributions using only sufficient statistics has long been recognized as a computationally hard problem. On the other hand, computationally efficient algorithms for learning Gibbs distributions rely on access to full sample configurations generated from the model. For many systems of interest that arise in physical contexts, expecting a full sample to be observed is not practical, and hence it is important to look for computationally efficient methods that solve the learning problem with access to only a limited set of statistics. We examine the trade-offs between the power of computation and observation within this scenario, employing the Ising model as a paradigmatic example. We demonstrate that it is feasible to reconstruct the model parameters for a model with $\ell_1$ width $γ$ by observing statistics up to an order of $O(γ)$. This approach allows us to infer the model's structure and also learn its couplings and magnetic fields. We also discuss a setting where prior information about structure of the model is available and show that the learning problem can be solved efficiently with even more limited observational power.
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