  T-distributed stochastic neighbour embedding (t-SNE) is a widely used data
visualisation technique. It differs from its predecessor SNE by the
low-dimensional similarity kernel: the Gaussian kernel was replaced by the
heavy-tailed Cauchy kernel, solving the "crowding problem" of SNE. Here, we
develop an efficient implementation of t-SNE for a $t$-distribution kernel with
an arbitrary degree of freedom $\nu$, with $\nu\to\infty$ corresponding to SNE
and $\nu=1$ corresponding to the standard t-SNE. Using theoretical analysis and
toy examples, we show that $\nu<1$ can further reduce the crowding problem and
reveal finer cluster structure that is invisible in standard t-SNE. We further
demonstrate the striking effect of heavier-tailed kernels on large real-life
data sets such as MNIST, single-cell RNA-sequencing data, and the HathiTrust
library. We use domain knowledge to confirm that the revealed clusters are
meaningful. Overall, we argue that modifying the tail heaviness of the t-SNE
kernel can yield additional insight into the cluster structure of the data.
