Clustering in the Boolean Hypercube in a List Decoding Regime

Abstract

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 $\Pr_{x\in X}[\left<{x,z}\right>\ge \varepsilon ]\ge \delta$, 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/polylog(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/n ^c, ε)-center for some constant c and ε = 1/poly(n), δ = 1/poly(n).