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Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints

James Cruickshank, Fatemeh Mohammadi, Anthony Nixon, Shin-ichi Tanigawa

TL;DR

The paper develops a general framework, called $g$-rigidity, to study identifiability of point configurations under algebraic constraints organized by $k$-uniform hypergraphs. It generalizes classical Euclidean rigidity by replacing distance with arbitrary polynomial measurements $g$, connecting identifiability to the geometry of secant varieties and their projections. Through eight concrete models (including ordinary Euclidean rigidity, tensor and matrix completions, and Chow decompositions), it derives combinatorial and algebraic criteria for local and global rigidity and provides tools (via Jacobians and infinitesimal motions) to analyze these problems. The work yields both theoretical insights into how combinatorics influence identifiability and practical implications for problems such as random projections of secant varieties and even even-$p$ norm rigidity, bridging rigidity theory, algebraic geometry, and tensor decomposition.

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.

Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints

TL;DR

The paper develops a general framework, called -rigidity, to study identifiability of point configurations under algebraic constraints organized by -uniform hypergraphs. It generalizes classical Euclidean rigidity by replacing distance with arbitrary polynomial measurements , connecting identifiability to the geometry of secant varieties and their projections. Through eight concrete models (including ordinary Euclidean rigidity, tensor and matrix completions, and Chow decompositions), it derives combinatorial and algebraic criteria for local and global rigidity and provides tools (via Jacobians and infinitesimal motions) to analyze these problems. The work yields both theoretical insights into how combinatorics influence identifiability and practical implications for problems such as random projections of secant varieties and even even- norm rigidity, bridging rigidity theory, algebraic geometry, and tensor decomposition.

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.
Paper Structure (16 sections, 4 theorems, 19 equations, 3 figures)

This paper contains 16 sections, 4 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rigidity of Hypergraphs
  3. Euclidean rigidity
  4. Rigidity under algebraic constraints
  5. Examples of $g$-rigidity Models
  6. Ordinary Euclidean rigidity.
  7. Rigidity in pseudo-Euclidean space.
  8. $\ell_p$-norm rigidity.
  9. Volume-constrained rigidity.
  10. Positive semidefinite symmetric matrix completion.
  11. Symmetric tensor completions.
  12. Skew-symmetric tensor completions.
  13. Chow decompositions.
  14. Tools for Analysing $g$-rigidity
  15. Basic facts on polynomial maps
  16. ...and 1 more sections

Key Result

Proposition 2.4

Suppose $V$ is a finite set and $p, q\in (\mathbb{R}^d)^{V}$. Then holds if and only if there is an Euclidean isometry $(A,t)$ such that $q(u)=Ap(u)+t\ (u\in V)$.

Figures (3)

  • Figure 1: Simple examples of frameworks in the Euclidean plane: (left) a flexible framework with motion indicated; and (right) a rigid framework.
  • Figure 2: (Left) An infinitesimal flex of the flexible framework in Figure \ref{['fig:flex']}. (Middle) A generically rigid graph whose edge set is dependent in the generic 2-dimensional rigidity matroid since $|E|=6>2|V|-3$. (Right) A generically rigid graph realised as a framework that has an infinitesimal flex (indicated).
  • Figure 3: (Left and middle) rigid but not globally rigid frameworks in the plane. (Right) a globally rigid framework in the plane.

Theorems & Definitions (12)

  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 4.1: Generic point
  • Proposition 4.2
  • proof
  • Definition 4.3
  • Proposition 4.4
  • proof
  • Example 4.5
  • ...and 2 more