# Semisupervised Clustering by Queries and Locally Encodable Source Coding

## TL;DR

This paper establishes an equivalence between semi-supervised clustering with simple queries and locally encodable source coding, deriving query complexity bounds and proposing efficient querying schemes with performance guarantees.

## Contribution

It introduces a novel connection between semi-supervised clustering and locally encodable source coding, providing lower bounds and practical querying strategies.

## Key findings

- Lower bounds on the number of queries needed for accurate clustering.
- Querying schemes based on pairwise 'same cluster' and AND queries.
- Provable performance guarantees for the proposed schemes.

## Abstract

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 semi-supervised 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 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 a variety of scenarios. We provide querying schemes based on pairwise `same cluster' queries - and pairwise AND queries and show provable performance guarantees for each of the schemes.