Homological algebra and data
Homological algebra and data
复制标题
同调代数和数据
DOI：
10.1090/pcms/025/06
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
R. Ghrist
中科院分区：
文献类型：
--
作者：
R. Ghrist
这些讲座是应用代数拓扑基础知识的快速入门，重点是数据的应用。特别地，（初等）同调代数的观点，以复形和共/同调不变量的形式被勾画。从简单和细胞复合体作为一种将图丰富到高阶结构的手段开始，我们定义了简单代数拓扑不变量，如欧拉特征。通过从简单复形提升到向量空间的代数复形，我们传递到同调作为一种拓扑压缩方案。迭代这个扩展到序列并通过同调代数压缩的过程，我们定义了持久同调和相关理论，并以一个简单的细胞束及其上同调的方法结束。在整个过程中，重点放在表达同调代数工具作为线性代数的自然演变。为了便于访问，不强调范畴论语言（虽然更自然、更有表现力）。在此过程中，概述了这些技术在从神经科学到传感、图像分析、机器人和计算等领域的示例应用。
These lectures are a quick primer on the basics of applied algebraic topology with emphasis on applications to data. In particular, the perspectives of (elementary) homological algebra, in the form of complexes and co/homological invariants are sketched. Beginning with simplicial and cell complexes as a means of enriching graphs to higher-order structures, we define simple algebraic topological invariants, such as Euler characteristic. By lifting from complexes of simplices to algebraic complexes of vector spaces, we pass to homology as a topological compression scheme. Iterating this process of expanding to sequences and compressing via homological algebra, we define persistent homology and related theories, ending with a simple approach to cellular sheaves and their cohomology. Throughout, an emphasis is placed on expressing homological-algebraic tools as the natural evolution of linear algebra. Category-theoretic language (though more natural and expressive) is deemphasized, for the sake of access. Along the way, sample applications of these techniques are sketched, in domains ranging from neuroscience to sensing, image analysis, robotics, and computation.
影响因子：
2.5
作者：
Singer, A.
通讯作者：
Singer, A.
影响因子：
0.5
作者：
Curry, J;Patel, A
通讯作者：
Patel, A