The maximum-entropy remote sampling problem (MERSP) is to select a subset of s random variables from a set of n random variables, so as to maximize the information concerning a set of target random variables that are not directly observable. We assume throughout that the set of all of these random variables follows a joint Gaussian distribution, and that we have the covariance matrix available. Finally, we measure information using Shannon's differential entropy. The main approach for exact solution of moderate-sized instances of MERSP has been branch-and-bound, and so previous work concentrated on upper bounds. Prior to our work, there were two upper-bounding methods for MERSP: the so-called NLP bound and the spectral bound, both introduced 25 years ago. We are able now to establish domination results between these two upper bounds. We propose an ``augmented NLP bound'' based on a subtle convex relaxation. We provide theoretical guarantees, giving sufficient conditions under which the augmented NLP bound strictly dominates the ordinary NLP bound. In addition, the augmented NLP formulation allows us to derive upper bounds for rank-deficient covariance matrices when they satisfy a technical condition. This is in contrast to the earlier work on the ordinary NLP bound that worked with only positive definite covariance matrices. Finally, we introduce a novel and very effective diagonal-scaling technique for MERSP, employing a positive vector of parameters. Numerical experiments on benchmark instances demonstrate the effectiveness of our approaches in advancing the state of the art for calculating upper bounds on MERSP.

翻译：最大熵远程采样问题（MERSP）旨在从n个随机变量中选取s个随机变量的子集，以最大化关于一组不可直接观测的目标随机变量的信息。我们始终假设所有这些随机变量服从联合高斯分布，且协方差矩阵已知。最后，我们使用香农微分熵作为信息度量。解决中等规模MERSP实例的主要精确方法是分支定界法，因此先前研究集中于上界计算。在我们工作之前，存在两种MERSP上界方法：即25年前提出的NLP界和谱界。我们现在能够建立这两种上界之间的支配关系。基于一种精妙的凸松弛方法，我们提出了“增强NLP界”。我们提供了理论保证，给出了增强NLP界严格支配普通NLP界的充分条件。此外，增强NLP公式允许我们在协方差矩阵满足特定技术条件时，推导秩亏协方差矩阵的上界。这与早期仅适用于正定协方差矩阵的普通NLP界研究形成对比。最后，我们引入了一种采用正参数向量的新型高效对角缩放技术。在基准实例上的数值实验表明，我们的方法在提升MERSP上界计算技术水平方面具有显著效果。