Clustering with t-SNE, Provably

t-distributed stochastic neighborhood embedding (t-SNE), a clustering and visualization method proposed by van der Maaten and Hinton in 2008, has rapidly become a standard tool in a number of natural sciences. Despite its overwhelming success, there is a distinct lack of mathematical foundations, and the inner workings of the algorithm are not well understood. The purpose of this paper is to prove that t-SNE is able to recover well-separated clusters; more precisely, we prove that t-SNE in the early exaggeration phase, an optimization technique proposed by van der Maaten and Hinton [J. Mach. Learn. Res., 9 (2008), pp. 2579-2605] and van der Maaten [J. Mach. Learn. Res., 15 (2014), pp. 3221-3245], can be rigorously analyzed. As a byproduct, the proof suggests novel ways for setting the exaggeration parameter alpha and step size h. Numerical examples illustrate the effectiveness of these rules: in particular, the quality of embedding of topological structures (e.g., the swiss roll) improves. We also discuss a connection to spectral clustering methods.