Statistical topology using persistence landscapes
Statistical topology using persistence landscapes
复制标题
使用持久性景观的统计拓扑
DOI：
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发表时间：
2012
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通讯作者：
Peter Bubenik
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作者：
Peter Bubenik
我们定义了一个新的描述符持久的同源性，我们称之为持久性景观，为了方便统计推断。该描述符可以被认为是将通常的描述符、条形码和持久性图嵌入到继承L范数的函数空间中。我们证明了相应的度量是拓扑等价的（p + 1）-Wasserstein距离，并且这个度量空间是完备的和可分的。我们证明了一个稳定性定理的持久景观。当p = 2时，我们证明了持久性景观的Fréchet均值是逐点均值，Fréchet方差是逐点方差的积分.此外，持久景观的样本平均值逐点收敛到底层分布的平均值，并且存在相应的中心极限定理。
We define a new descriptor for persistent homology, which we call the persistence landscape, for the purpose of facilitating statistical inference. This descriptor may be thought of as an embedding of the usual descriptors, barcodes and persistence diagrams, into a space of functions, which inherits an L norm. We show that the corresponding metric is topologically equivalent to the (p + 1)-Wasserstein distance, and that this metric space is complete and separable. We prove a stability theorem for persistence landscapes. For p = 2, we show that the Fréchet mean of persistence landscapes is the pointwise mean, and that the Fréchet variance is the integral of the pointwise variances. Furthermore, the sample mean of persistence landscapes converges pointwise to the mean of the underlying distribution, and there is a corresponding central limit theorem.