Clustering in the boolean hypercube in a list decoding regime

• Authors: Irit Dinur;Elazar Goldenberg
• Venue: ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
• Year: 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 ≥ 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/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/nc, ε)-center for some constant c and ε=1/poly(n), δ=1/poly(n).