Sensing with Optimal Matrices

TL;DR

This paper addresses the design of optimal sensing matrices that minimize the worst-case estimation error by reducing the maximum condition number of all submatrices, providing explicit solutions for specific cases.

Contribution

It derives explicit formulas and optimal matrix designs for M=2, K=3, and arbitrary N, revealing that uniform distributions are not always optimal.

Findings

Optimal matrices for M=2, K=3, arbitrary N are derived.

Explicit formulas for the condition number of submatrices are provided.

Uniform distributions are often suboptimal for minimizing maximum condition number.

Abstract

We consider the problem of designing optimal $M \times N$ ($M \leq N$) sensing matrices which minimize the maximum condition number of all the submatrices of $K$ columns. Such matrices minimize the worst-case estimation errors when only $K$ sensors out of $N$ sensors are available for sensing at a given time. For M=2 and matrices with unit-normed columns, this problem is equivalent to the problem of maximizing the minimum singular value among all the submatrices of $K$ columns. For M=2, we are able to give a closed form formula for the condition number of the submatrices. When M=2 and K=3, for an arbitrary $N\geq3$, we derive the optimal matrices which minimize the maximum condition number of all the submatrices of $K$ columns. Surprisingly, a uniformly distributed design is often \emph{not} the optimal design minimizing the maximum condition number.