K-median: exact recovery in the extended stochastic ball model

TL;DR

This paper investigates the conditions under which the k-median LP achieves exact recovery in the stochastic ball model and its extension, providing new theoretical insights and a generalized data model.

Contribution

It corrects previous tightness results, establishes new tightness conditions in high and low dimensions, and introduces the extended stochastic ball model with exact recovery guarantees.

Findings

k-median LP is not tight in low dimensions

tighter conditions for high-dimensional cases

extended stochastic ball model allows exact recovery

Abstract

We study exact recovery conditions for the linear programming relaxation of the k-median problem in the stochastic ball model (SBM). In Awasthi et al. (2015), the authors give a tight result for the k-median LP in the SBM, saying that exact recovery can be achieved as long as the balls are pairwise disjoint. We give a counterexample to their result, thereby showing that the k-median LP is not tight in low dimension. Instead, we give a near optimal result showing that the k-median LP in the SBM is tight in high dimension. We also show that, if the probability measure satisfies some concentration assumptions, then the k-median LP in the SBM is tight in every dimension. Furthermore, we propose a new model of data called extended stochastic ball model (ESBM), which significantly generalizes the well-known SBM. We then show that exact recovery can still be achieved in the ESBM.