  Dimensionality reduction is a common method for analyzing and visualizing
high-dimensional data. However, reasoning dynamically about the results of a
dimensionality reduction is difficult. Dimensionality-reduction algorithms use
complex optimizations to reduce the number of dimensions of a dataset, but
these new dimensions often lack a clear relation to the initial data
dimensions, thus making them difficult to interpret. Here we propose a visual
interaction framework to improve dimensionality-reduction based exploratory
data analysis. We introduce two interaction techniques, forward projection and
backward projection, for dynamically reasoning about dimensionally reduced
data. We also contribute two visualization techniques, prolines and feasibility
maps, to facilitate the effective use of the proposed interactions. We apply
our framework to PCA and autoencoder-based dimensionality reductions. Through
data-exploration examples, we demonstrate how our visual interactions can
improve the use of dimensionality reduction in exploratory data analysis.
