Problem 46: Fast JL Transform for Sparse Vectors
Revision as of 04:14, 17 November 2012 by Krzysztof Onak (talk | contribs) (Created page with "{{Header |title=Fast JL Transform for Sparse Vectors |source=bertinoro11 |who=Jelani Nelson }} Consider a distribution over linear mappings $A$ from $R^d$ to $R^k$, $k=O(\log...")
| Suggested by | Jelani Nelson |
|---|---|
| Source | Bertinoro 2011 |
| Short link | https://sublinear.info/46 |
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]).
Question: Provide an explicit construction of a distribution with the $(\epsilon, P)$-JL property such that the random variable $A$ can be generated using $O(\log (d/P))$ bits.
