title: A graph-based N-body approximation with application to stochastic neighbor embedding.
abstract: We propose a novel approximation technique, bubble approximation (BA), for repulsion forces in an N-body problem, where attraction has a limited range and repulsion acts between all points. These kinds of systems occur frequently in dimension reduction and graph drawing. Like tree codes, the established N-body approximation method, BA replaces several point-to-point computations by one area-to-point computation. Novelty of BA is to consider not only the magnitudes but also the directions of forces from the area. Therefore, its area-to-point approximations are applicable anywhere in the space. The joint effect of forces from inside the area is calculated analytically, assuming a homogeneous mass of points inside the area. These two features free BA from hierarchical data structures and complicated bookkeeping of interactions, which plague tree codes. Instead, BA uses a simple graph to control the computations. The graph provides a sparse matrix, which, suitably weighted, replaces the full matrix of pairwise comparisons in the N-body problem. As a concrete example, we implement a sparse-matrix version of stochastic neighbor embedding (a dimension reduction method), and demonstrate its good performance by comparisons to full-matrix method, and to three different approximate versions of the same method.
authors.affiliation: NBE, Aalto University School of Science, PO Box 12200, FI-00076, Finland. Electronic address: eli.parviainen@iki.fi.
authors.forename: Eli
authors.initials: E
authors.lastname: Parviainen
mesh_terms: D006801:Humans; D000069550:Machine Learning; D016571:Neural Networks, Computer; D013269:Stochastic Processes
keywords: Approximation method; Dimension reduction; Machine learning; N-body problem; Stochastic neighbor embedding