CONSISTENT MANIFOLD REPRESENTATION FOR TOPOLOGICAL DATA ANALYSIS
CONSISTENT MANIFOLD REPRESENTATION FOR TOPOLOGICAL DATA ANALYSIS
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DOI：
10.3934/fods.2019001
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发表时间：
2019-03-01
影响因子：
2.3
通讯作者：
Sauer, Timothy
中科院分区：
文献类型：
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作者：
Berry, Tyrus;Sauer, Timothy
对于嵌入欧几里德空间的流形上任意密度采样的数据，引入连续k近邻（CkNN）图构造。证明了CkNN在几何上是一致的，即在一定条件下，非归一化图拉普拉斯算子收敛于拉普拉斯-贝尔特拉米算子，在谱上和点上都是一致的。对于紧致流形证明（对于非紧致流形推测），CkNN是唯一的非加权结构，在大数据的限制下，它产生与底层流形的连接组件一致的几何形状。因此，CkNN产生一个同时捕获所有拓扑特征的单个图，而不是持久同调，它在单独的尺度上表示每个同调生成器。作为应用，我们提出了一种新的快速聚类算法和一种从拓扑上识别自然图像模式的方法。最后，我们推测CkNN是拓扑一致的，这意味着在大数据的极限下，Vietoris-Rips复合体的同调（由图拉普拉斯算子隐含）收敛于底层流形的同调（由拉普拉斯-德拉姆算子隐含）。
For data sampled from an arbitrary density on a manifold embedded in Euclidean space, the Continuous k-Nearest Neighbors (CkNN) graph construction is introduced. It is shown that CkNN is geometrically consistent in the sense that under certain conditions, the unnormalized graph Laplacian converges to the Laplace-Beltrami operator, spectrally as well as pointwise. It is proved for compact (and conjectured for noncompact) manifolds that CkNN is the unique unweighted construction that yields a geometry consistent with the connected components of the underlying manifold in the limit of large data. Thus CkNN produces a single graph that captures all topological features simultaneously, in contrast to persistent homology, which represents each homology generator at a separate scale. As applications we derive a new fast clustering algorithm and a method to identify patterns in natural images topologically. Finally, we conjecture that CkNN is topologically consistent, meaning that the homology of the Vietoris-Rips complex (implied by the graph Laplacian) converges to the homology of the underlying manifold (implied by the Laplace-de Rham operators) in the limit of large data.