Theory Seminar: Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

Speaker:
Gil Cohen (Weizmann Institute of Science)
Date:
Wednesday, 25.6.2014, 12:30
Place:
Taub 201

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n, there exists an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball of equal volume embedded in $(n+1)$-dimensional Boolean cube, such that for all $x$ and $y$ it holds that $\mathrm{distance}(x,y) / 5 \leq \mathrm{distance}(f(x),f(y)) \leq 4 \mathrm{distance}(x,y)$, where $\mathrm{distance}(,)$ denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive.

This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who raised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions requires ideas beyond the sensitivity-based structural results of Boppana [IPL 97].

No prior knowledge is assumed.

Joint work with Itai Benjamini and Igor Shinkar.

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