High-dimensional inference with the generalized Hopfield model: Principal component analysis and corrections
High-dimensional inference with the generalized Hopfield model: Principal component analysis and corrections
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DOI：
10.1103/physreve.83.051123
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发表时间：
2011-05-20
影响因子：
2.4
通讯作者：
Sessak, V.
中科院分区：
文献类型：
--
作者：
Cocco, S.;Monasson, R.;Sessak, V.
我们考虑的问题推断一组N个二进制变量之间的相互作用，从他们的频率和成对的相关性的知识。推理框架基于Hopfield模型，这是Ising模型的一种特殊情况，其中交互矩阵通过变量空间中的一组模式来定义，并且其秩远小于N。我们表明，最大似然推断是深相关的主成分分析时，模式分量的振幅是可以忽略不计的V根N。使用统计力学的技术，我们计算了对模式的修正，以xi/root N为一阶。我们强调需要推广的Hopfield模型，包括吸引力和排斥力的模式，以正确地推断网络稀疏和强大的相互作用。我们提出了一个简单的几何准则来决定有多少吸引力和排斥模式应被视为采样噪声的函数。此外，我们讨论了一个好的推理需要多少采样配置，作为系统大小N和幅度xi的函数。合成和生物数据的推理方法说明。
We consider the problem of inferring the interactions between a set of N binary variables from the knowledge of their frequencies and pairwise correlations. The inference framework is based on the Hopfield model, a special case of the Ising model where the interaction matrix is defined through a set of patterns in the variable space, and is of rank much smaller than N. We show that maximum likelihood inference is deeply related to principal component analysis when the amplitude of the pattern components xi is negligible compared to v root N. Using techniques from statistical mechanics, we calculate the corrections to the patterns to the first order in xi/root N. We stress the need to generalize the Hopfield model and include both attractive and repulsive patterns in order to correctly infer networks with sparse and strong interactions. We present a simple geometrical criterion to decide how many attractive and repulsive patterns should be considered as a function of the sampling noise. We moreover discuss how many sampled configurations are required for a good inference, as a function of the system size N and of the amplitude xi. The inference approach is illustrated on synthetic and biological data.