## 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.

Original language | English |
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Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Neural Networks |

Volume | 75 |

DOIs | |

Publication status | Published - 1 Mar 2016 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Approximation method
- Dimension reduction
- Machine learning
- N-body problem
- Stochastic neighbor embedding