Homework 2 Foundations of Algorithms for Massive Datasets ( 236779 ) Fall 2015
نویسنده
چکیده
Small coherence (or, incoherence) has nice properties as we shall see now. Assume A has coherence μ < 1. Prove that for all s < 1/μ, A has RIP with parameters s, δ = sμ. Hint: For s-sparse x, write down ‖Ax‖ = (Ax)Ax using inner products of columns of A. Note: There exist deterministic constructions of k×d matrices with incoherence ∼ 1/ √ k, which by the above imply RIP with parameters s = √ k, δ = O(1). But we know how to obtain, using random constructions (eg JL, as you did above) matrices of the same size and with RIP of much better parameters s = k/ log d, δ = O(1). This shows a gap between what we can achieve using deterministic vs randomized constructions. For certain applications, incoherence might be enough.
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