Overlap Removal of Dimensionality Reduction Scatterplot Layouts

Dimensionality Reduction (DR) scatterplot layouts have become a ubiquitous visualization tool for analyzing multidimensional data items with presence in different areas. Despite its popularity, scatterplots suffer from occlusion, especially when markers convey information, making it troublesome for users to estimate items' groups' sizes and, more importantly, potentially obfuscating critical items for the analysis under execution. Different strategies have been devised to address this issue, either producing overlap-free layouts, lacking the powerful capabilities of contemporary DR techniques in uncover interesting data patterns, or eliminating overlaps as a post-processing strategy. Despite the good results of post-processing techniques, the best methods typically expand or distort the scatterplot area, thus reducing markers' size (sometimes) to unreadable dimensions, defeating the purpose of removing overlaps. This paper presents a novel post-processing strategy to remove DR layouts' overlaps that faithfully preserves the original layout's characteristics and markers' sizes. We show that the proposed strategy surpasses the state-of-the-art in overlap removal through an extensive comparative evaluation considering multiple different metrics while it is 2 or 3 orders of magnitude faster for large datasets.Comment: 11 pages and 9 figure

Simultaneous Embeddability of Two Partitions

We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201

Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference