138 research outputs found
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
An Incidence Geometry approach to Dictionary Learning
We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a
sparse representation of data points, by learning \emph{dictionary vectors}
upon which the data points can be written as sparse linear combinations. We
view this problem from a geometry perspective as the spanning set of a subspace
arrangement, and focus on understanding the case when the underlying hypergraph
of the subspace arrangement is specified. For this Fitted Dictionary Learning
problem, we completely characterize the combinatorics of the associated
subspace arrangements (i.e.\ their underlying hypergraphs). Specifically, a
combinatorial rigidity-type theorem is proven for a type of geometric incidence
system. The theorem characterizes the hypergraphs of subspace arrangements that
generically yield (a) at least one dictionary (b) a locally unique dictionary
(i.e.\ at most a finite number of isolated dictionaries) of the specified size.
We are unaware of prior application of combinatorial rigidity techniques in the
setting of Dictionary Learning, or even in machine learning. We also provide a
systematic classification of problems related to Dictionary Learning together
with various algorithms, their assumptions and performance
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
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