Covering point-sets with parallel hyperplanes and sparse signal recovery
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Let S be a set of k > n points in a Euclidean space R^n, n ≥ 1. How many parallel hyperplanes are needed to cover the set S? We prove that any such set can covered by k-n+1 hyperplanes and construct examples of sets that cannot be covered by fewer parallel hyperplanes. We then demonstrate a construction of a family of n × d integer matrices from difference vectors of such point-sets, d ≥ n, with bounded sup-norm and the property that no ℓ column vectors are linearly dependent, ℓ≤ n. Further, if ℓ≤ (log n)^1-ε for any ε > 0, then d/n →∞ as n →∞. This is an explicit construction of a family of sensing matrices, which are used for sparse recovery of integer-valued signals in compressed sensing.
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