Masking Anstreicher's linx bound for improved entropy bounds

TL;DR

This paper demonstrates that masking can significantly improve Anstreicher's linx bound for the maximum-entropy sampling problem, providing a linear improvement in the upper bound even with optimal scaling.

Contribution

The authors show that masking can enhance the linx bound for the maximum-entropy sampling problem, establishing a linear improvement with optimal scaling.

Findings

Masking improves the linx bound by at least a linear amount in n.

The improvement holds even with optimal scaling parameters.

The method applies to covariance matrices in spatial statistics.

Abstract

The maximum-entropy sampling problem is the NP-hard problem of maximizing the (log) determinant of an order-$s$ principle submatrix of a given order $n$ covariance matrix $C$. Exact algorithms are based on a branch-and-bound framework. The problem has wide applicability in spatial statistics, and in particular in environmental monitoring. Probably the best upper bound for the maximum, empirically, is Anstreicher's scaled ``linx'' bound (see [K.M. Anstreicher. Efficient solution of maximum-entropy sampling problems. Oper. Res., 68(6):1826--1835, 2020]). An earlier methodology for potentially improving any upper-bounding method is by masking; i.e. applying the bounding method to $C\circ M$, where $M$ is any correlation matrix. We establish that the linx bound can be improved via masking by an amount that is at least linear in $n$, even when optimal scaling parameters are employed.