A statistical physics approach to learning curves for the inverse Ising problem
A statistical physics approach to learning curves for the inverse Ising problem
复制标题
伊辛逆问题学习曲线的统计物理方法
DOI：
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发表时间：
2017
期刊：
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通讯作者：
M. Opper
中科院分区：
文献类型：
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作者：
Ludovica Bachschmid;M. Opper
利用统计物理的方法，分析了从独立数据(伊辛逆问题)学习大伊辛模型耦合的误差。我们专注于基于局部成本函数的学习，例如伪似然方法，对于该方法，耦合是针对每个自旋独立地推断的。假定数据是从真实的伊辛模型中产生的，对于密集连接系统，我们使用复制法和腔方法相结合的方法来计算耦合的重建误差。我们证明了基于二次代价函数的显式估计器可以获得最小的重建误差，但需要将真实耦合向量的长度作为先验知识。不需要这种知识的耦合的简单平均场估计器是渐近最优的，即当观测数目远远大于自旋数目时。理论和数值模拟的比较表明，在高温区有随机耦合的两个模型产生的数据非常符合：一个是具有独立耦合的模型(Sherrington-Kirkpatrick模型)，另一个是耦合矩阵具有Wishart分布的模型。
Using methods of statistical physics, we analyse the error of learning couplings in large Ising models from independent data (the inverse Ising problem). We concentrate on learning based on local cost functions, such as the pseudo-likelihood method for which the couplings are inferred independently for each spin. Assuming that the data are generated from a true Ising model, we compute the reconstruction error of the couplings using a combination of the replica method with the cavity approach for densely connected systems. We show that an explicit estimator based on a quadratic cost function achieves minimal reconstruction error, but requires the length of the true coupling vector as prior knowledge. A simple mean field estimator of the couplings which does not need such knowledge is asymptotically optimal, i.e. when the number of observations is much larger than the number of spins. Comparison of the theory with numerical simulations shows excellent agreement for data generated from two models with random couplings in the high temperature region: a model with independent couplings (Sherrington–Kirkpatrick model), and a model where the matrix of couplings has a Wishart distribution.