Recovery a planted signal perturbed by noise is a fundamental problem in machine learning. In this work, we consider the problem of recovery a planted $k$-densest sub-hypergraph on $h$-uniform hypergraphs over $n$ nodes. This fundamental problem appears in different contexts, e.g., community detection, average case complexity, and neuroscience applications. We first observe that it can be viewed as a structural variant of tensor PCA in which the hypergraph parameters $k$ and $h$ determine the structure of the signal to be recovered when the observations are contaminated by Gaussian noise. In this work, we provide tight information-theoretic upper and lower bounds for the recovery problem, as well as the first non-trivial algorithmic bounds based on approximate message passing algorithms. The problem exhibits a typical information-to-computational-gap observed in analogous settings, that widens with increasing sparsity of the problem. Interestingly, the bounds show that the structure of the signal does have an impact on the existing bounds of tensor PCA that the unstructured planted signal does not capture.
翻译:在这项工作中,我们考虑的是对美元单式高压高压高压高压高压高压高压高压电压的人工加压子高压速谱进行回收的问题。这一根本问题出现在不同的背景中,例如社区探测、平均案件复杂度和神经科学应用等。我们首先发现,它可被视为高压五氯苯甲醚的一个结构变体,其中高压参数为美元和美元,确定在观测受到高斯噪音污染时要恢复的信号结构。在这项工作中,我们为恢复问题提供了紧密的信息理论上下界,以及基于大致信息传递算法的第一个非三维算法界限。问题显示了在类似环境下观察到的典型信息到算法,随着问题日益紧张而扩大。有趣的是,这些界限表明信号的结构确实对未结构化的信号无法捕捉捉到的索诺尔常设仲裁院的现有界限产生了影响。