Clustering in the Boolean Hypercube in a List Decoding Regime.
Irit Dinur, Elazar Goldenberg · 2013
We consider the following clustering with outliers problem: Given a set of points X ⊂ {−1, 1}n, such that there is some point z ∈ {−1, 1}n for which Prx∈X [〈x, z 〉 ≥ ε] ≥ δ, find z. We call such a point z a (δ, ε)-center of X. In this work we give lower and upper bounds for the task of finding a (δ, ε)-center. We first show that for δ = 1−ν close to 1, i.e. in the unique decoding regime, given a (1−ν, ε)-centered set our algorithm can find a (1−(1+o(1))ν, (1−o(1))ε)-center. More interestingly, we study the list decoding regime, i.e. when δ is close to 0. Our main upper bound shows that for values of ε and δ that are larger than 1/poly log(n), there exists a polynomial time algorithm that finds a (δ − o(1), ε − o(1))-center. Moreover, our algorithm outputs a list of centers explaining all of the clusters in the input. Our main lower bound shows that given a set for which there exists a (δ, ε)-center, it is hard to find even a (δ/nc, ε)-center for some constant c and ε = 1/poly(n), δ = 1/poly(n).