Semisupervised Clustering, AND-Queries and Locally Encodable Source Coding

Source coding is the canonical problem of data compression in information theory. In a locally encodable source coding, each compressed bit depends on only few bits of the input. In this paper, we show that a recently popular model of semisupervised clustering is equivalent to locally encodable source coding. In this model, the task is to perform multiclass labeling of unlabeled elements. At the beginning, we can ask in parallel a set of simple queries to an oracle who provides (possibly erroneous) binary answers to the queries. The queries cannot involve more than two (or a fixed constant number $\Delta$ of) elements. Now the labeling of all the elements (or clustering) must be performed based on the (noisy) query answers. The goal is to recover all the correct labelings while minimizing the number of such queries. The equivalence to locally encodable source codes leads us to find lower bounds on the number of queries required in variety of scenarios. We are also able to show fundamental limitations of pairwise `same cluster' queries - and propose pairwise AND queries, that provably performs better in many situations.