Explicit Dimension Reduction and Its Applications

• Authors: Zohar S. Karnin;Yuval Rabani;Amir Shpilka
• Venue: SIAM Journal on Computing
• Year: 2012

We construct a small set of explicit linear transformations mapping $\mathbb{R}^n$ to $\mathbb{R}^t$, where $t=O(\log (\gamma^{-1}) \epsilon^{-2})$, such that the $L_2$ norm of any vector in $\mathbb{R}^n$ is distorted by at most $1\pm \epsilon$ in at least a fraction of $1 - \gamma$ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is suboptimal compared with probabilistic arguments, we nevertheless are able to apply our construction to a number of problems. In particular, we use it to construct an $\epsilon$-sample (or pseudorandom generator) for linear threshold functions on $\mathbb{S}^{n-1}$ for $\epsilon = o(1)$. We also use it to construct an $\epsilon$-sample for spherical digons in $\mathbb{S}^{n-1}$ for $\epsilon = o(1)$. This construction leads to an efficient oblivious derandomization of the Goemans-Williamson Max-Cut algorithm and similar approximation algorithms (i.e., we construct a small set of hyperplanes such that for any instance we can choose one of them to generate a good solution). Our technique for constructing an $\epsilon$-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on $k$-wise independent sample spaces.