Recovery thresholds in the sparse planted matching problem

The planted $k$-factor problem

We consider the problem of recovering an unknown $k$-factor, hidden in a weighted random graph. For $k=1$ this is the planted matching problem, while the $k=2$ case is closely related to the planted travelling salesman problem. The inference problem is solved by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted sub-graph. We argue that, in the large size limit, a phase transition can appear between a full and a partial recovery phase as function of the signal-to-noise ratio. We give a criterion for the loc…

Planted matching problems on random hypergraphs

We consider the problem of inferring a matching hidden in a weighted random $k$-hypergraph. We assume that the hyperedges' weights are random and distributed according to two different densities conditioning on the fact that they belong to the hidden matching, or not. We show that, for $k>2$ and in the large graph size limit, an algorithmic first order transition in the signal strength separates a regime in which a complete recovery of the hidden matching is feasible from a regime in which partial recovery is possible. This is in contrast to the $k=2$ case where the transition is known to be c…

Parallel Tempering for the planted clique problem

The theoretical information threshold for the planted clique problem is $2\log_2(N)$, however no polynomial algorithm is known to recover a planted clique of size $O(N^{1/2-ε})$, $ε>0$. In this paper we will apply a standard method for the analysis of disordered models, the Parallel-Tempering (PT) algorithm, to the clique problem, showing numerically that its time-scaling in the hard region is indeed polynomial for the analyzed sizes. We also apply PT to a different but connected model, the Sparse Planted Independent Set problem. In this situation thresholds should be sharper and finite size c…

稀疏近似问题：快速模拟退火的成功与失败

Sparse approximation problem: how rapid simulated annealing succeeds and fails

基于稀疏性的信息处理技术已经在多个学科中得到了积极的研究。其中，通过过完备基的少量基向量的组合来近似表示给定数据集的数学框架被称为{ \em稀疏近似}。在本文中，我们应用模拟退火，一般优化问题的元启发式算法，稀疏近似的情况下，给定的数据有一个种植稀疏表示和噪声存在。在无噪声情况下的结果表明，我们的模拟退火工程以及在一个合理的参数区域：种植的解决方案是相当迅速地找到。即使在稀疏近似问题的常见松弛（$\ell_1$ -松弛）无效的情况下也是如此。另一方面，当数据的维数接近非零分量的数量时，另一个亚稳态出现，我们的算法无法找到种植解决方案。这种现象与一阶相变有关。在非常强的噪声的情况下，寻找种植的解决方案不再有意义。在这种情况下，我们的算法可以相当快地确定接近最小失真的解决方案。

Mismatching as a tool to enhance algorithmic performances of Monte Carlo methods for the planted clique model

Over-parametrization was a crucial ingredient for recent developments in inference and machine-learning fields. However a good theory explaining this success is still lacking. In this paper we study a very simple case of mismatched over-parametrized algorithm applied to one of the most studied inference problem: the planted clique problem. We analyze a Monte Carlo (MC) algorithm in the same class of the famous Jerrum algorithm. We show how this MC algorithm is in general suboptimal for the recovery of the planted clique. We show however how to enhance its performances by adding a (mismatched) …

带结构化信号的贝叶斯推断：种植自旋玻璃西村线上的静态复制对称破缺

Bayesian Inference with Structured Signal: Static Replica Symmetry Breaking on the Nishimori Line in the Planted Spin Glass

该研究针对随机正则图上的种植自旋玻璃模型，分析信号结构对贝叶斯推断的影响，发现西村条件下先验的复制对称破缺相可引发后验的静态复制对称破缺，并探讨其对置信传播算法性能的影响。