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arXiv · 1206.0277

Sensing with Optimal Matrices

Hema Kumari Achanta·Soura Dasgupta·Weiyu Xu
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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.

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Hema Kumari Achanta, Soura Dasgupta, Weiyu Xu. 2012-06-01. Sensing with Optimal Matrices. https://arxiv.org/abs/1206.0277

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