基于密度度量学习的内在持久同源性
Intrinsic persistent homology via density-based metric learning
摘要：
我们解决的问题估计的拓扑特征，从数据在高维欧氏空间下的流形假设。我们的方法是基于持久的同源性的数据点赋予一个样本度量称为费马距离的空间的计算。我们证明，这样的度量空间收敛几乎肯定的流形本身赋予了一个内在的度量，占的几何形状的流形和密度，产生的样本。这一事实暗示了相关持久性图的收敛。当计算持久同源性时，使用这种固有距离呈现有利的性质，例如对输入数据中存在离群值的鲁棒性和对周围空间中底层流形的特定嵌入的不敏感性。我们使用这些想法，提出并实现了一种方法，模式识别和异常检测的时间序列，这是在实际数据的应用程序进行评估。
英文摘要：
We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with a sample metric known as Fermat distance. We prove that such metric space converges almost surely to the manifold itself endowed with an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact implies the convergence of the associated persistence diagrams. The use of this intrinsic distance when computing persistent homology presents advantageous properties such as robustness to the presence of outliers in the input data and less sensitiveness to the particular embedding of the underlying manifold in the ambient space. We use these ideas to propose and implement a method for pattern recognition and anomaly detection in time series, which is evaluated in applications to real data.
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