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arXiv · 1410.1826

Almost Separable Matrices

Matthew Aldridge·Leonardo Baldassini·Karen Gunderson
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Abstract

An $m \times n$ matrix $\mathsf{A}$ with column supports $\{S_i\}$ is $k$-separable if the disjunctions $\bigcup_{i \in \mathcal{K}} S_i$ are all distinct over all sets $\mathcal{K}$ of cardinality $k$. While a simple counting bound shows that $m > k \log_2 n/k$ rows are required for a separable matrix to exist, in fact it is necessary for $m$ to be about a factor of $k$ more than this. In this paper, we consider a weaker definition of `almost $k$-separability', which requires that the disjunctions are `mostly distinct'. We show using a random construction that these matrices exist with $m = O(k \log n)$ rows, which is optimal for $k = O(n^{1-\beta})$. Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing.

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Matthew Aldridge, Leonardo Baldassini, Karen Gunderson. 2014-10-07. Almost Separable Matrices. https://doi.org/10.1007/s10878-015-9951-1

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