稀疏神经在实践中

摘要：

拓扑数据分析将机器学习与代数拓扑方法相结合。持续同源性，一种方法来描述发生在多个尺度的数据的拓扑特征是特别感兴趣的。持久同源性广泛应用的一个主要障碍是它的计算复杂性。为了能够计算大型数据集的持久同源性，可以应用许多近似以降低其复杂性。我们提出的算法计算近似稀疏神经类Dowker相异，包括所有有限Dowker相异和Dowker相异的同源性是切赫持续同源性。我们所知道的所有其他稀疏化方法和软件包都使用加法或乘法交织来计算持久同源性。在dowker_homology中，我们允许任何非减交织函数$α$。我们分析了这些算法的计算复杂性，并给出了一些基准测试。对于大于3维的欧几里得数据，我们创建的单纯复形的大小通常比SimBa创建的小小于。特别是当在更高的同源性维度中计算持久同源性时，差异可能变得很大。

英文摘要：

Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker dissimilarities and Dowker dissimilarities whose homology is Cech persistent homology. All other sparsification methods and software packages that we are aware of calculate persistent homology with either an additive or a multiplicative interleaving. In dowker_homology, we allow for any non-decreasing interleaving function $α$. We analyze the computational complexity of the algorithms and present some benchmarks. For Euclidean data in dimensions larger than three, the sizes of simplicial complexes we create are in general smaller than the ones created by SimBa. Especially when calculating persistent homology in higher homology dimensions, the differences can become substantial.

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