Visualizing distortions and recovering topology in continuous projection techniques

The visualization of continuous multi-dimensional data based on their projection to a 2-dimensional space is a way to detect visually interesting patterns, as far as the projection provides a faithful image of the original data. In order to evaluate this faithfulness, we propose to visualize any measure associated to a projected datum or to a pair of projected data, by coloring the corresponding Voronoï cell in the projection space. We also define specific measures and show how they allow estimating visually whether some part of the projection is or is not a reliable image of the original manifolds. It also helps to figure out what the original topology of the data is, telling where the high-dimensional manifolds have been torn or glued during the projection. We experiment these techniques with the principal component analysis and the curvilinear component analysis applied to artificial and real databases.