发表机构
Lancaster University（兰卡斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于最近邻的降维方法，该方法的矩阵是密度信息矩阵的一致估计量，可辅助聚类分析与异常检测等下游任务。
AI 中文摘要
本文提出一种直观的降维方法，能高效识别多元数据的有效投影，其核心思路与现有多项技术类似，旨在增强数据中的最近邻关系。该方法的投影来自一个矩阵的谱分解，此矩阵用于编码数据的局部协方差结构，而某点的局部协方差由其最近邻对捕获。研究表明，在标准正则条件下，该矩阵是所谓“密度信息矩阵（DIM）”的一致估计量，DIM是费舍尔信息矩阵的非参数类似物。已有研究证实，DIM的谱分解与独立成分分析、监督场景下的充分降维等重要问题相关。不过，现有DIM估计量计算成本高昂，且仅针对替代密度的DIM，该替代密度与真实基础密度的平方成正比。此外，本文还进一步探索了该方法在辅助聚类分析、异常检测等下游任务中的实际应用价值。
英文摘要
An intuitive method for dimensionality reduction is proposed, which is highly effective for finding interesting projections of multivariate data. Following similar intuitive motivation to a number of existing techniques, the proposed method is based on enhancing the nearest neighbour relationships in the data. The proposed projection arises from the spectral decomposition of a matrix designed to encode the local covariance structure in the data, where the local covariance at a point is captured by pairs of its nearest neighbours. We show that under standard regularity conditions this matrix is a consistent estimator of the so-called ``Density Information Matrix'' (DIM); a non-parametric analogue of the Fisher Information Matrix. Spectral decompositions of DIMs have been shown to be connected with the important problems of Independent Components Analysis and, in the supervised context, Sufficient Dimension Reduction. However, existing estimators of the DIM are computationally expensive to compute and only target the DIM of a surrogate density, which is proportional to the square of the true underlying density. In addition, we go on to explore the practical utility of our method in aiding the downstream tasks of cluster analysis and outlier detection.
