We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, which corresponds to a non-convex, non-smooth optimisation problem. We show that the optimal univariate projection based on spectral connectivity converges to the vector normal to the maximum margin hyperplane through the data, as the scaling parameter is reduced to zero. This establishes a connection between connectivity as measured by spectral graph theory and maximal Euclidean separation. The computational cost associated with each eigen-problem is quadratic in the number of data. To mitigate this issue, we propose an approximation method using microclusters with provable approximation error bounds. Combining multiple binary partitions within a divisive hierarchical model allows us to construct clustering solutions admitting clusters with varying scales and lying within different subspaces. We evaluate the performance of the proposed method on a large collection of benchmark datasets and find that it compares favourably with existing methods for projection pursuit and dimension reduction for data clustering.
Optimal Projections for Classification with Naive Bayes
David P. Hofmeyr School of Mathematical Sciences, Lancaster University, United Kingdom（大卫·P·霍夫梅尔 威廉·朗斯特大学数学科学学院）；Francois Kamper Swiss Data Science Centre, EPFL & ETH Zürich, Switzerland（弗朗索瓦·卡默尔 瑞士数据科学中心，EPFL与苏黎世联邦理工学院）；Michail C. Melonas Kohort, Cape Town, South Africa（米哈伊尔·C·梅洛纳斯 哥特，开普敦，南非）
MarmoNet: a pipeline for automated projection mapping of the common marmoset brain from whole-brain serial two-photon tomography
RIKEN Center for Brain Science（理化学研究所脑科学中心）；Keio University School of Medicine（庆应义塾大学医学院）；Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）；Kyoto University（京都大学）；Janelia Farm Research Campus（珍妮莉亚农场研究园区）
