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Peer-reviewedCC BYJournal of the London Mathematical Society

Identifiability of points and rigidity of hypergraphs under algebraic constraints

  • James G. Cruickshank1
  • Fatemeh Mohammadi2
  • Anthony Nixon3
  • Shin‐ichi Tanigawa4
4 affiliations
  1. 1School of Mathematical and Statistical Sciences University of Galway Galway Ireland
  2. 2Departments of Mathematics and Computer Science KU Leuven Leuven Belgium
  3. 3School of Mathematical Sciences Lancaster University Lancaster UK
  4. 4Department of Business Economics Tokyo University of Science Tokyo Japan
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@article{cruickshank2026identifiability,
  title = {{Identifiability of points and rigidity of hypergraphs under algebraic constraints}},
  author = {James G. Cruickshank and Fatemeh Mohammadi and Anthony Nixon and Shin‐ichi Tanigawa},
  journal = {Journal of the London Mathematical Society},
  year = {2026},
  month = jul,
  doi = {10.1112/jlms.70633},
  url = {https://doi.org/10.1112/jlms.70633},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Identifiability of points and rigidity of hypergraphs under algebraic constraints
AU  - James G. Cruickshank
AU  - Fatemeh Mohammadi
AU  - Anthony Nixon
AU  - Shin‐ichi Tanigawa
PY  - 2026
DA  - 2026/07/01
JO  - Journal of the London Mathematical Society
DO  - 10.1112/jlms.70633
UR  - https://doi.org/10.1112/jlms.70633
AB  - 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.
SN  - 0024-6107
SN  - 1469-7750
ER  -