Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints
CoRR(2023)
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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Key words
hypergraphs,rigidity
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