Many statistical inference problems correspond to recovering the values of a set of hidden variables from sparse observations on them. For instance, in a planted constraint satisfaction problem such as planted 3-SAT, the clauses are sparse observations from which the hidden assignment is to be recovered. In the problem of community detection in a stochastic block model, the community labels are hidden variables that are to be recovered from the edges of the graph. Inspired by ideas from statistical physics, the presence of a stable fixed point for belief propogation has been widely conjectured to characterize the computational tractability of these problems. For community detection in stochastic block models, many of these predictions have been rigorously confirmed. In this work, we consider a general model of statistical inference problems that includes both community detection in stochastic block models, and all planted constraint satisfaction problems as special cases. We carry out the cavity method calculations from statistical physics to compute the regime of parameters where detection and recovery should be algorithmically tractable. At precisely the predicted tractable regime, we give: (i) a general polynomial-time algorithm for the problem of detection: distinguishing an input with a planted signal from one without; (ii) a general polynomial-time algorithm for the problem of recovery: outputting a vector that correlates with the hidden assignment significantly better than a random guess would.
翻译:许多统计推论问题与从对一组隐藏变量的零散观测中恢复一套隐藏变量的价值相对应。例如,在种植的3SAT等植入的限制性满意度问题中,条款是很少的观察,从中可以找到隐藏的任务。在一个随机区块模型中,社区标签是隐藏的变量,从图边缘中可以找到。在统计物理思想的启发下,信仰传播的稳定固定点的存在被广泛推断为这些问题的可计算性特征。对于在随机区块模型中社区探测的问题,许多这些预测已得到严格确认。在这项工作中,我们考虑统计推断问题的一般模式,包括社区在随机区块模型中的检测,以及所有植入的制约满意度问题作为特殊情况。我们从统计物理学中进行隐蔽方法的计算,以比较参数制度,在其中检测和恢复应具有逻辑性可测量性。精确地说,我们给出了:(一)一般的多元-时间算法,许多预测区块模型已经得到严格确认。我们考虑了统计问题的一般模型模型模型的精确性算算法,而没有对一个精确的模型进行精确的定量测测算。