Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints
Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints
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DOI：
10.48550/arxiv.2305.18990
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发表时间：
2023-05
期刊：
影响因子：
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通讯作者：
J. Cruickshank;F. Mohammadi;A. Nixon;Shin-ichi Tanigawa
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文献类型：
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作者：
J. Cruickshank;F. Mohammadi;A. Nixon;Shin-ichi Tanigawa
可识别性问题在数学和计算机科学的许多背景下自然出现。具体的例子包括图的局部或全局刚性和部分填充张量在秩条件下的唯一完备性。割线簇上的点的可识别性也是代数几何中的一个研究课题。它通常被公式化为识别满足给定代数关系集的点集的问题。一个关键的问题，然后是证明充分条件的关系，以保证可识别的点。本文提出了一个新的一般框架捕获的可识别性问题时，一组代数关系具有组合结构，并开发工具来分析潜在的组合对局部或全局可识别性的点的影响。我们的框架是建立在语言的图形刚性，其中的测量是两点之间的欧几里得距离，但适用于任意代数测量的超图的一般性。我们建立必要和充分的（超）图论的可识别性条件，利用技术从图的刚性理论和代数几何割线品种。特别是我们的工作组合分析割线品种的非通用预测的效果。
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.