Topological data analysis provides a multiscale description of the geometry
and topology of quantitative data. The persistence landscape is a topological
summary that can be easily combined with tools from statistics and machine
learning. We give efficient algorithms for calculating persistence landscapes,
their averages, and distances between such averages. We discuss an
implementation of these algorithms and some related procedures. These are
intended to facilitate the combination of statistics and machine learning with
topological data analysis. We present an experiment showing that the
low-dimensional persistence landscapes of points sampled from spheres (and
boxes) of varying dimensions differ.

用拓扑数据分析方法，通过计算保持景观等拓扑信息的特征，将不同维度的几何数据转化为低纬度拓扑信息，并将其与统计学和机器学习方法相结合，以便从中提取有用的信息。

用于拓扑统计的持久景观工具箱

A persistence landscapes toolbox for topological statistics

Persistent homology captures the evolution of topological features of a model
as a parameter changes. The most commonly used summary statistics of persistent
homology are the barcode and the persistence diagram. Another summary
statistic, the persistence landscape, was recently introduced by Bubenik. It is
a functional summary, so it is easy to calculate sample means and variances,
and it is straightforward to construct various test statistics. Implementing a
permutation test we detect conformational changes between closed and open forms
of the maltose-binding protein, a large biomolecule consisting of 370 amino
acid residues. Furthermore, persistence landscapes can be applied to machine
learning methods. A hyperplane from a support vector machine shows the clear
separation between the closed and open proteins conformations. Moreover,
because our approach captures dynamical properties of the protein our results
may help in identifying residues susceptible to ligand binding; we show that
the majority of active site residues and allosteric pathway residues are
located in the vicinity of the most persistent loop in the corresponding
filtered Vietoris-Rips complex. This finding was not observed in the classical
anisotropic network model.

采用持久同调的方法对大分子进行构象变化的分析，并利用持久景观方法进行测试和分类，结果表明这种方法能够成功的识别出分子中活性位点和趋肤通路。

利用持续同调和动态距离分析蛋白质结合

Using persistent homology and dynamical distances to analyze protein  binding

Persistent homology is a widely used tool in Topological Data Analysis that
encodes multiscale topological information as a multi-set of points in the
plane called a persistence diagram. It is difficult to apply statistical theory
directly to a random sample of diagrams. Instead, we can summarize the
persistent homology with the persistence landscape, introduced by Bubenik,
which converts a diagram into a well-behaved real-valued function. We
investigate the statistical properties of landscapes, such as weak convergence
of the average landscapes and convergence of the bootstrap. In addition, we
introduce an alternate functional summary of persistent homology, which we call
the silhouette, and derive an analogous statistical theory.

该研究论文介绍了多尺度拓扑信息编码和持久拓扑学中的统计推论及其应用，其中总结持久性景观和轮廓，对它们的平均值进行弱收敛和启发式收敛的研究，并推导出持久拓扑学的一个函数总结。

持久景观和轮廓的随机收敛

Stochastic Convergence of Persistence Landscapes and Silhouettes

We define a new topological summary for data that we call the persistence
landscape. Since this summary lies in a vector space, it is easy to combine
with tools from statistics and machine learning, in contrast to the standard
topological summaries. Viewed as a random variable with values in a Banach
space, this summary obeys a strong law of large numbers and a central limit
theorem. We show how a number of standard statistical tests can be used for
statistical inference using this summary. We also prove that this summary is
stable and that it can be used to provide lower bounds for the bottleneck and
Wasserstein distances.

本篇研究提出一种叫做 “持久化景观” 的数据拓朴表示方法，可以方便地和统计学和机器学习相结合，并可以用于统计学推断和类距离的下界估计。