Exploring High-Dimensional Structure via Axis-Aligned Decomposition of Linear Projections

Two-dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non-linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis-aligned projections, which are easy to interpret but are limited to bivariate relationships. Linear project can be considered as a compromise between complexity and interpretability, as they allow explicit axes labels, yet provide significantly more degrees of freedom compared to axis-aligned projections. Nevertheless, interpreting the axes directions, which are linear combinations often with many non-trivial components, remains difficult. To address this problem we introduce a structure aware decomposition of (multiple) linear projections into sparse sets of axis aligned projections, which jointly capture all information of the original linear ones. In particular, we use tools from Dempster-Shafer theory to formally define how relevant a given axis aligned project is to explain the neighborhood relations displayed in some linear projection. Furthermore, we introduce a new approach to discover a diverse set of high quality linear projections and show that in practice the information of $k$ linear projections is often jointly encoded in $\sim k$ axis aligned plots. We have integrated these ideas into an interactive visualization system that allows users to jointly browse both linear projections and their axis aligned representatives. Using a number of case studies we show how the resulting plots lead to more intuitive visualizations and new insight.