How to sample if you must: on optimal functional sampling

TL;DR

This paper investigates the problem of optimally selecting linear functionals for sampling a multivariate normal distribution to minimize estimation error, providing bounds and an efficient sub-optimal solution for structured functional sets.

Contribution

It introduces bounds and a practical sub-optimal method for optimal functional sampling, especially for structured sets like graph-induced binary functionals.

Findings

Derived bounds for optimal sampling strategies.

Developed an efficient sub-optimal sampling algorithm.

Applicable to structured sets such as graph walks.

Abstract

We examine a fundamental problem that models various active sampling setups, such as network tomography. We analyze sampling of a multivariate normal distribution with an unknown expectation that needs to be estimated: in our setup it is possible to sample the distribution from a given set of linear functionals, and the difficulty addressed is how to optimally select the combinations to achieve low estimation error. Although this problem is in the heart of the field of optimal design, no efficient solutions for the case with many functionals exist. We present some bounds and an efficient sub-optimal solution for this problem for more structured sets such as binary functionals that are induced by graph walks.