  Data visualization is one of the major applications of nonlinear
dimensionality reduction. From the information retrieval perspective, the
quality of a visualization can be evaluated by considering the extent that the
neighborhood relation of each data point is maintained while the number of
unrelated points that are retrieved is minimized. This property can be
quantified as a trade-off between the mean precision and mean recall of the
visualization. While there have been some approaches to formulate the
visualization objective directly as a weighted sum of the precision and recall,
there is no systematic way to determine the optimal trade-off between these two
nor a clear interpretation of the optimal value. In this paper, we investigate
the properties of $\alpha$-divergence for information visualization, focusing
our attention on a particular range of $\alpha$ values. We show that the
minimization of the new cost function corresponds to maximizing a geometric
mean between precision and recall, parameterized by $\alpha$. Contrary to some
earlier methods, no hand-tuning is needed, but we can rigorously estimate the
optimal value of $\alpha$ for a given input data. For this, we provide a
statistical framework using a novel distribution called Exponential Divergence
with Augmentation (EDA). By the extensive set of experiments, we show that the
optimal value of $\alpha$, obtained by EDA corresponds to the optimal trade-off
between the precision and recall for a given data distribution.
