Recent developments in shape reconstruction and comparison call for the use of many different types of topological descriptors (persistence diagrams, Euler characteristic functions, etc.). We establish a framework that allows for quantitative comparisons of topological descriptor types and therefore may be used as a tool in more rigorously justifying choices made in applications. We then use this framework to partially order a set of six common topological descriptor types. In particular, the resulting poset gives insight into the advantages of using verbose rather than concise topological descriptors. We then provide lower bounds on the size of sets of descriptors that are complete discrete invariants of simplicial complexes, both tight and worst case. This work sets up a rigorous theory that allows for future comparisons and analysis of topological descriptor types.

翻译：近年来，形状重建与比较领域的发展需要运用多种不同类型的拓扑描述符（如持续同调图、欧拉特征函数等）。我们建立了一个框架，能够对拓扑描述符类型进行定量比较，从而为应用中的选择提供更严格的依据。利用该框架，我们对六种常见拓扑描述符类型进行了偏序排列。值得注意的是，由此得到的偏序集揭示了采用详尽型而非简洁型拓扑描述符的优势。随后，我们给出了作为单纯复形完备离散不变量所需的描述符集合大小的下界，既包含紧确界也包含最坏情况。本研究构建了严谨的理论体系，为未来拓扑描述符类型的比较与分析奠定了基础。