Problem 46: Fast JL Transform for Sparse Vectors

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Suggested by Jelani Nelson
Source Bertinoro 2011
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Consider a distribution over linear mappings $A$ from $R^d$ to $R^k$, $k=O(\log (1/P)/\epsilon^2)$. We say that it has an $(\epsilon, P)$-JL property if for any vector $x \in R^d$ we have $$\|Ax\|_2 = (1 \pm \epsilon) \|x\|_2$$ with probability $1-P$.

Question: Can we construct a distribution with this property such that the matrix-vector product $Ax$ can be evaluated in time $(s+k)\cdot \operatorname{polylog}(d)$ time given an $s$-sparse $x$?

Background: Such an algorithm is not known even for $s=d$ (unless $k$ is larger [AilonL-11], [KrahmerW-11]).

Question: Provide an explicit construction of a distribution with the $(\epsilon, P)$-JL property such that the random matrix $A$ can be generated using $O(\log (d/P))$ bits.