论文标题
度量空间幅度和加权向量的实际应用
Practical applications of metric space magnitude and weighting vectors
论文作者
论文摘要
公制空间幅度是代数拓扑研究的积极研究主题，最初是在生物学的背景下出现的，在生物学的背景下，它被用来表示环境中有效数量的不同物种。在更通用的环境中，度量空间的大小是一个实数，旨在量化空间中有效数量的有效数量。每个点对由{\ em加权向量}编码的度量空间的全局幅度的贡献，捕获了原始度量空间的许多基本几何形状。 令人惊讶的是，当度量空间是欧几里得时，加权矢量也是边界检测的有效工具。这使加权向量可以作为经典机器学习任务（例如分类，离群检测和主动学习）的新算法的基础。我们证明，使用经典基准数据集上的实验和比较，这是提出的幅度和基于权重矢量方法的承诺。
Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting, the magnitude of a metric space is a real number that aims to quantify the effective number of distinct points in the space. The contribution of each point to a metric space's global magnitude, which is encoded by the {\em weighting vector}, captures much of the underlying geometry of the original metric space. Surprisingly, when the metric space is Euclidean, the weighting vector also serves as an effective tool for boundary detection. This allows the weighting vector to serve as the foundation of novel algorithms for classic machine learning tasks such as classification, outlier detection and active learning. We demonstrate, using experiments and comparisons on classic benchmark datasets, the promise of the proposed magnitude and weighting vector-based approaches.
