Lower bounds for oblivious subspace embeddings
Abstract
An oblivious subspace embedding (OSE) for some eps, delta in (0,1/3) and d <= m <= n is a distribution D over Rm x n such that for any linear subspace W of Rn of dimension d, PrPi ~ D(for all x in W, (1-eps) |x|2 <= |Pi x|2 <= (1+eps)|x|2) >= 1 - delta. We prove that any OSE with delta < 1/3 must have m = Omega((d + log(1/delta))/eps2), which is optimal. Furthermore, if every Pi in the support of D is sparse, having at most s non-zero entries per column, then we show tradeoff lower bounds between m and s.
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