Detecting structure in data is the first step to arrive at meaningful
representations for systems. This is particularly challenging for dislocation
networks evolving as a consequence of plastic deformation of crystalline
systems. Our study employs Isomap, a manifold learning technique, to unveil the
intrinsic structure of high-dimensional density field data of dislocation
structures from different compression axis. The resulting maps provide a
systematic framework for quantitatively comparing dislocation structures,
offering unique fingerprints based on density fields. Our novel, unbiased
approach contributes to the quantitative classification of dislocation
structures which can be systematically extended.

利用 Isomap 技术揭示不同压缩轴下位阶密度场数据的位错结构的内在结构，并提供了基于密度场的独特指纹，这种新颖的无偏方法有助于位错结构的定量分类。

位错制图：具有独特指纹的位错网络的表示和无监督分类

Dislocation cartography: Representations and unsupervised classification  of dislocation networks with unique fingerprints

Multidimensional Scaling (MDS) is one of the first fundamental manifold
learning methods. It can be categorized into several methods, i.e., classical
MDS, kernel classical MDS, metric MDS, and non-metric MDS. Sammon mapping and
Isomap can be considered as special cases of metric MDS and kernel classical
MDS, respectively. In this tutorial and survey paper, we review the theory of
MDS, Sammon mapping, and Isomap in detail. We explain all the mentioned
categories of MDS. Then, Sammon mapping, Isomap, and kernel Isomap are
explained. Out-of-sample embedding for MDS and Isomap using eigenfunctions and
kernel mapping are introduced. Then, Nystrom approximation and its use in
landmark MDS and landmark Isomap are introduced for big data embedding. We also
provide some simulations for illustrating the embedding by these methods.

本文介绍了多维尺度变换（MDS）及其不同类型、Sammon 映射、Isomap 等算法，以及使用特征函数和核映射等方法处理大数据的 out-of-sample MDS 和 Isomap 嵌入，并着重介绍了 Nystrom 近似及其在地标 MDS 和 Isomap 中的应用。

多维度缩放、Sammon 映射和 Isomap：教程与综述

Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and  Survey

Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and
Laplacian Eigenmaps address the problem of representing high-dimensional
nonlinear data in terms of low-dimensional coordinates which represent the
intrinsic structure of the data. This paradigm incorporates the assumption that
real-valued coordinates provide a rich enough class of functions to represent
the data faithfully and efficiently. On the other hand, there are simple
structures which challenge this assumption: the circle, for example, is
one-dimensional but its faithful representation requires two real coordinates.
In this work, we present a strategy for constructing circle-valued functions on
a statistical data set. We develop a machinery of persistent cohomology to
identify candidates for significant circle-structures in the data, and we use
harmonic smoothing and integration to obtain the circle-valued coordinate
functions themselves. We suggest that this enriched class of coordinate
functions permits a precise NLDR analysis of a broader range of realistic data
sets.

本文提出了利用统计数据集构建圆形值函数的方法，并使用持久性上同调的机器来识别数据中显著的圆形结构，通过谐波平滑和积分得到圆值坐标函数，进一步拓宽了坐标函数的类别，以更精确地进行 NLDR 分析。