This week, I visited Afonso Bandeira at NYU to give a talk in the MaD seminar on the semidefinite relaxation of k-means. Here are the slides. The last part of the talk is very new; I worked it out with Soledad Villar while she visited me a couple weeks ago, and our paper just hit the arXiv. In this blog entry, I’ll briefly summarize the main idea of the paper.

Suppose you are given data points , and you are tasked with finding the partition that minimizes the k-means objective

(Here, we normalize the objective by for convenience later.) To do this, you will likely run MATLAB’s built-in implementation of k-means++, which randomly selects of the data points (with an intelligent choice of random distribution), and then uses these data points as proto-centroids to initialize Lloyd’s algorithm. In practice, this works very well: After running it a few times, you generally get a very nice clustering. But when do you know to stop looking for an even better clustering?

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