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Explicit Dimension Reduction and its Applications
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Zohar Karnin, Ph.D. Thesis Seminar
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Wednesday, 25.5.2011, 14:30
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Taub 601
We construct a small set of explicit linear transformations mapping R^n to R^t, where t=O(log(\gamma^{-1}) \epsilon^{-2}), such that the L_2 norm of any vector in 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 sub-optimal compated with probablistic 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 pseudo-random-generator) for linear threshold functions on S^{n-1}, for \epsilon=o(1). We also use it to construct an \epsilon-sample for digons in 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.
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