伊辛模型的计算充分统计量
Computationally sufficient statistics for Ising models
AI总结：
本文研究了在仅访问有限统计量的情况下，如何通过伊辛模型高效学习参数及耦合项，展示了计算能力和观测权衡的可行方法。
摘要：
利用仅有的充分统计量来学习吉布斯分布长期以来被认为是一个计算上具有挑战性的问题。另一方面，用于学习吉布斯分布的计算高效算法依赖于能够访问来自模型的完整样本配置。对于许多在物理背景下出现的系统，期望获得完整的样本是不实际的，因此寻找能够在仅访问有限统计量的情况下解决学习问题的计算高效方法就显得很重要。我们在此情境下考察了计算能力和观测之间的权衡，以伊辛模型作为典范例子。我们证明了通过观测到阶为O(γ)的统计量，可以重建具有ℓ₁宽度为γ的模型参数。这种方法使我们能够推断模型的结构，并且学习其耦合项和磁矩。我们还讨论了一种情况，其中关于模型结构的先验信息是可用的，并展示了即使在更有限的观测能力下，学习问题也可以被高效地解决。
英文摘要：
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 show that the couplings of the model can be learned to a desired accuracy from statistics whose order grows linearly with the $\ell_1$ width of the model and only logarithmically with the inverse of the accuracy. In particular, for a fixed accuracy, the required order is proportional to the width. This allows us to infer the model's structure, using statistics whose order grows logarithmically with the inverse of the weakest coupling, and also to learn its magnetic fields. When the width and the accuracy are fixed, the running time is polynomial in the number of variables. 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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