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Random projections in the Euclidean space reduce the dimensionality of the data approximately preserving the distances between points. In the hypercube it holds a weaker property: random projections approximately preserve the distances within a certain range.
In this note, we show an analogous result for the metric space 〈Σd,dH〉, where Σd is the set of words of length d on alphabet Σ and dH is the Hamming distance.
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