Sparse Hypergraphs and Pebble Game Algorithms

A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V′⊂V spans more than k|V′|−ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs

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

Graded Sparse Graphs and Matroids

Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically pinned (completely immobilized) bar-and-joint frameworks and prove that they also form matroids. We address five problems on graded sparse graphs: {\bf Decision}, {\bf Extraction}, {\bf Components}, {\bf Optimization}, and {\bf Extension}. We extend our {\bf pebble game algorithms} to solve them.Comment: 9 pages, 1 figure; improved presentation and fixed typo

Natural realizations of sparsity matroids

A hypergraph G with n vertices and m hyperedges with d endpoints each is (k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a linearly representable matroidal family. Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence structure of the underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars Mathematica Contemporane

Analyzing Rigidity with Pebble Games

How many pair-wise distances must be prescribed between an unknown set of points, and how should they be distributed, to determine only a discrete set of possible solutions? These questions, and related generalizations, are central in a variety of applications. Combinatorial rigidity shows that in two-dimensions one can get the answer, generically, via an efficiently testable sparse graph property. We present a video and a web site illustrating algorithmic results for a variety of rigidity-related problems, as well as abstract generalizations. Our accompanying interactive software is based on a comprehensive implementation of the pebble game paradigm

Body-and-cad geometric constraint systems

AbstractMotivated by constraint-based CAD software, we develop the foundation for the rigidity theory of a very general model: the body-and-cad structure, composed of rigid bodies in 3D constrained by pairwise coincidence, angular and distance constraints. We identify 21 relevant geometric constraints and develop the corresponding infinitesimal rigidity theory for these structures. The classical body-and-bar rigidity model can be viewed as a body-and-cad structure that uses only one constraint from this new class.As a consequence, we identify a new, necessary, but not sufficient, counting condition for minimal rigidity of body-and-cad structures: nested sparsity. This is a slight generalization of the well-known sparsity condition of Maxwell