The identifiability problem arises naturally in a number of contexts in mathematics and computer science. Specific instances include local or global rigidity of graphs and unique completability of partially-filled tensors subject to rank conditions. The identifiability of points on secant varieties has also been a topic of much research in algebraic geometry. It is often formulated as the problem of identifying a set of points satisfying a given set of algebraic relations. A key question then is to prove sufficient conditions for relations 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 analyse 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. In particular our work analyses combinatorially the effect of non-generic projections of secant varieties.

翻译：可识别性问题在数学和计算机科学的多个领域中自然产生，具体实例包括图的局部或全局刚性以及满足秩条件的部分填充张量的唯一完备性。割线簇上点的可识别性也一直是代数几何中的研究热点，通常被表述为识别满足给定代数关系集的一组点的问题。其中的关键问题是证明确保这些关系能保证点可识别性的充分条件。本文提出了一种新的通用框架，用于描述当代数关系集具有组合结构时的可识别性问题，并开发了分析底层组合结构对点局部或全局可识别性影响的工具。该框架基于图刚性的语言构建（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情景。我们通过利用图刚性理论及割线簇代数几何的技术，建立了（超）图论意义上可识别性的充要条件。特别地，本研究从组合角度分析了割线簇非一般投影的影响。