Explicit Dimension Reduction and Its Applications

We construct a small set of explicit linear transformations mapping ℝⁿ to ℝ^t, where t = O(log(γ^-1)ϵ^-2), such that the L₂ norm of any vector in ℝⁿ is distorted by at most 1±ϵ in at least a fraction of 1-γ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is sub-optimal 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 ϵ-sample (or pseudo-random generator) for linear threshold functions on S^n-1, for ϵ = o(1). We also use it to construct an ϵ-sample for spherical digons in S^n-1, for ϵ = o(1). This construction leads to an efficient oblivious derandomization of the Goemans-Williamson MAXCUT 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 ϵ-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on k-wise independent sample spaces.