Clustering in the Boolean Hypercube in a List Decoding Regime

We consider the following clustering with outliers problem: Given a set of points X ⊂ {−1, 1}, such that there is some point z ∈ {−1, 1} for which Prx∈X [〈x, z〉 ≥ ε] ≥ δ, nd 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 nding a (δ, ε)-center. 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 nds a (δ− o(1), ε− o(1))-center. Moreover, it 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 nd even a (δ/n, ε)-center for some constant c and ε = 1/poly(n), δ = 1/poly(n). ∗Weizmann Institute of Science and Radcli e Institute for Advanced Study. Research supported in part by the Israel Science Foundation grant no. 1179/09 and by the Binational Science Foundation grant no. 2008293 and by an ERC grant no. 239985. †Weizmann Institute of Science.