Identifiability of data is one of the fundamental problems in data science. Mathematically it is often formulated as the identifiability of points satisfying a given set of algebraic relations. A key question then is to identify sufficient conditions for observations to guarantee the identifiability of the points. This paper proposes a new general framework for capturing the identifiability problem when a set of algebraic relations has a combinatorial structure and develops tools to analyze the impact of the underlying combinatorics on the local or global identifiability of points. Our framework is built on the language of graph rigidity, where the measurements are Euclidean distances between two points, but applicable in the generality of hypergraphs with arbitrary algebraic measurements. We establish necessary and sufficient (hyper)graph theoretical conditions for identifiability by exploiting techniques from graph rigidity theory and algebraic geometry of secant varieties.

翻译：数据的可辨识性是数据科学中的基本问题之一。在数学上，这通常被形式化为满足给定代数关系集点的可辨识性。那么，一个关键问题是确定确保点可辨识性的观测充分条件。本文提出了一种新的通用框架，用于刻画当代数关系集具有组合结构时的可辨识性问题，并开发了分析底层组合结构对点的局部或全局可辨识性影响的工具。我们的框架建立在图刚性的语言之上（其中测量值为两点间的欧氏距离），但可推广至具有任意代数测量值的超图。通过利用图刚性理论及割线簇的代数几何技术，我们建立了可辨识性的（超）图理论充要条件。