114 research outputs found
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
Rigidity through a Projective Lens
In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar−joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body−hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas
The orbit rigidity matrix of a symmetric framework
A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks. The orbit matrix is a true
analog of the standard rigidity matrix for general frameworks, and its analysis
gives important insights into questions about the flexibility and rigidity of
classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the
power to predict symmetry-preserving finite mechanisms - giving a simplified
analysis which covers a wide range of the known mechanisms, and generalizes the
classes of known mechanisms. This initial exploration of the properties of the
orbit matrix also opens up a number of new questions and possible extensions of
the previous results, including transfer of symmetry based results from
Euclidean space to spherical, hyperbolic, and some other metrics with shared
symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure
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