Data quality is crucial for the successful training, generalization and
performance of artificial intelligence models. Furthermore, it is known that
the leading approaches in artificial intelligence are notoriously data-hungry.
In this paper, we propose the use of small training datasets towards faster
training. Specifically, we provide a novel topological method based on
morphisms between persistence modules to measure the training data quality with
respect to the complete dataset. This way, we can provide an explanation of why
the chosen training dataset will lead to poor performance.

本文提出了一种基于持续模块之间的同态的新拓扑方法，以衡量训练数据的质量，并解释为什么所选的训练数据集将导致性能不佳，从而提出使用小型训练数据集来加快训练。

衡量训练数据质量的拓扑方法

A Topological Approach to Measuring Training Data Quality

This paper is a survey of persistent homology, primarily as it is used in
topological data analysis. It includes the theory of persistence modules, as
well as stability theorems for persistence barcodes, generalized persistence,
vectorization of persistence barcodes, as well as some applications.

该文章是对持久同调的调查，主要是它在拓扑数据分析中的应用，包括持久模的理论、持久条形码的稳定性定理、广义持久性、持久条形码的向量化，以及一些应用。

持久同调和应用同伦理论

Persistent Homology and Applied Homotopy Theory

We give a self-contained treatment of the theory of persistence modules
indexed over the real line. We give new proofs of the standard results.
Persistence diagrams are constructed using measure theory. Linear algebra
lemmas are simplified using a new notation for calculations on quiver
representations. We show that the stringent finiteness conditions required by
traditional methods are not necessary to prove the existence and stability of
the persistence diagram. We introduce weaker hypotheses for taming persistence
modules, which are met in practice and are strong enough for the theory still
to work. The constructions and proofs enabled by our framework are, we claim,
cleaner and simpler.

本文通过使用新符号计算 quiver representations 和测度理论构造 Persistence diagrams, 提出弱假设可以替代传统的强假设来证明 Persistence diagram 的存在和稳定性。我们的框架具有更加简洁和干净的性质。

持久模型的结构和稳定性

The structure and stability of persistence modules

We consider sequences of absolute and relative homology and cohomology groups
that arise naturally for a filtered cell complex. We establish algebraic
relationships between their persistence modules, and show that they contain
equivalent information. We explain how one can use the existing algorithm for
persistent homology to process any of the four modules, and relate it to a
recently introduced persistent cohomology algorithm. We present experimental
evidence for the practical efficiency of the latter algorithm.

我们研究了一种分级胞复合体的绝对与相对同调和余同调群的序列。我们建立了它们持久模之间的代数关系，并展示了它们包含相同的信息。我们解释了如何使用现有的持久同调算法处理其中任意一种模块，并将其与最近引入的持久余同调算法联系起来。我们提供了后一算法的实际效率的实验证据。