持久同调与持久同调:综述

Busayo Adeyege Okediji, A. Oladipo
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引用次数: 0

摘要

持久同调是非线性数据还原的重要工具。它的姊妹理论--持久同调在过去几年里受到的关注较少,尽管它有很多优点。我们对涉及持久同调与同调理论和计算的一些文献进行了调查。找出了同调学长期以来被忽视的原因,并提出了解决这些问题的一些可能方案。此外,还利用 Ripserer,以手工和计算两种方式计算了 2 球的持久同调和同调。在这两种情况下,都得到了相同的结果,特别是在计算它们的条形码时,明显发现了两者的重合点。最后,我们发现持久同调不仅计算速度比持久同调快,而且占用内存少。
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Persistent Homology and Persistent Cohomology: A Review
Persistent homology is an important tool in non-linear data reduction. Its sister theory, persistent cohomology, has attracted less attention in the past years eventhough it has many advantages. Several literatures dealing with theory and computations of persistent homology and cohomology were surveyed. Reasons why cohomology has been neglected over time are identified and, few possible solutions to the identified problems are made available. Furthermore, using Ripserer, the computation of persistent homology and cohomology using 2-sphere both manually and computationally are carried out. In both cases, same result was obtained, particularly in the computation of their barcodes which visibly revealed the point where the two coincides. Conclusively, it is observed that persistent cohomology is not only faster in computation than persistent homology, but also uses less memory in a little time.
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