Almost separable matrices

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.