Sketch-and-solve approaches to k-means clustering by semidefinite programming

One of the most fundamental data processing tasks is clustering. Here, one is given a collection of objects, a notion of similarity between those objects, and a clustering objective that scores any given partition according to how well it clusters similar objects together. The goal is then to partition the objects in a way that optimizes this clustering objective. For example, to partition the vertices of a simple graph into two clusters, one might take the clustering objective to be edge cut in the graph complement. This clustering problem is equivalent to MAX-CUT, which is known to be NP-hard [21].