Matrix Design for Optimal Sensing

TL;DR

This paper presents a method for designing 2xN matrices with unit columns that minimize the worst-case condition number among all 3-column submatrices, optimizing 2D signal estimation and sensor placement.

Contribution

It derives the optimal matrix configurations for any N≥3, revealing that uniform column distribution is not always optimal, especially for odd N≥7.

Findings

Optimal matrices minimize maximum condition number of 3-column submatrices.

Uniform distribution of columns is not optimal for odd N≥7.

Provides explicit solutions for all N≥3.

Abstract

We design optimal $2 \times N$ ($2 <N$) matrices, with unit columns, so that the maximum condition number of all the submatrices comprising 3 columns is minimized. The problem has two applications. When estimating a 2-dimensional signal by using only three of $N$ observations at a given time, this minimizes the worst-case achievable estimation error. It also captures the problem of optimum sensor placement for monitoring a source located in a plane, when only a minimum number of required sensors are active at any given time. For arbitrary $N\geq3$, we derive the optimal matrices which minimize the maximum condition number of all the submatrices of three columns. Surprisingly, a uniform distribution of the columns is \emph{not} the optimal design for odd $N\geq 7$.