Sampling Hypergraphs with Given Degrees

There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm $\mathcal{A}$ for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm $\mathcal{A}$, and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a constant

Sampling uniform hypergraphs with given degrees

Graphs are combinatorial objects commonly used to model relationships between pairs of entities. Hypergraphs are a generalization of graphs in which edges connect an arbitrary number of vertices. We consider hypergraphs in which each edge has size k, each vertex has a degree specified by a degree sequence d, and all edges are unique. These are known as simple k-uniform hypergraphs with degree sequence d. We focus on algorithms for sampling these hypergraphs, particularly when the degree sequence is approximately regular and sufficiently sparse. The goal is an algorithm which produces approximately uniform output with expected running time that is polynomial in the number of vertices. We first discuss an algorithm for this problem which used a rejection sampling approach and a black-box bipartite graph sampler. This algorithm was presented in a paper by myself and co-authors: my specific contributions to the publication are described. As a new contribution (not contained in the paper), the rejection sampling approach is extended to give an algorithm for sampling linear hypergraphs, which are hypergraphs in which no two distinct edges share more than one common vertex. We also define and analyse an algorithm for sampling simple k-uniform hypergraphs with degree sequence d. Our algorithm uses a black-box sampler A for producing (possibly non-simple) hypergraphs and a ‘switchings’ process to remove any repeated edges from the hypergraph. This analysis additionally produces explicit tail bounds for the number and multiplicity of repeated edges in uniformly distributed random hypergraphs, under certain conditions for d and k. We show that our algorithm is asymptotically approximately uniform and has an expected running time that is polynomial in the number of vertices for a large range of degree sequences d, provided d is near-regular. This extends the range of degree sequences for which efficient sampling schemes are known

Hypergraph Modelling for Geometric Model Fitting

In this paper, we propose a novel hypergraph based method (called HF) to fit and segment multi-structural data. The proposed HF formulates the geometric model fitting problem as a hypergraph partition problem based on a novel hypergraph model. In the hypergraph model, vertices represent data points and hyperedges denote model hypotheses. The hypergraph, with large and "data-determined" degrees of hyperedges, can express the complex relationships between model hypotheses and data points. In addition, we develop a robust hypergraph partition algorithm to detect sub-hypergraphs for model fitting. HF can effectively and efficiently estimate the number of, and the parameters of, model instances in multi-structural data heavily corrupted with outliers simultaneously. Experimental results show the advantages of the proposed method over previous methods on both synthetic data and real images.Comment: Pattern Recognition, 201

Toric algebra of hypergraphs

The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in algebraic statistics and to combinatorial discrepancy. Section 6 (open problems) has been moderately revise

New Notions and Constructions of Sparsification for Graphs and Hypergraphs

A sparsifier of a graph $G$ (Bencz\'ur and Karger; Spielman and Teng) is a sparse weighted subgraph $\tilde G$ that approximately retains the cut structure of $G$. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers, there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of edges in each cut $(S,\bar S)$ is not approximated within a multiplicative factor $(1+\epsilon)$, but is, instead, approximated up to an additive term bounded by $\epsilon$ times $d\cdot |S| + \text{vol}(S)$, where $d$ is the average degree, and $\text{vol}(S)$ is the sum of the degrees of the vertices in $S$. We provide a probabilistic polynomial time construction of such sparsifiers for every graph, and our sparsifiers have a near-optimal number of edges $O(\epsilon^{-2} n {\rm polylog}(1/\epsilon))$. We also provide a deterministic polynomial time construction that constructs sparsifiers with a weaker property having the optimal number of edges $O(\epsilon^{-2} n)$. Our constructions also satisfy a spectral version of the ``additive sparsification'' property. Our construction of ``additive sparsifiers'' with $O_\epsilon (n)$ edges also works for hypergraphs, and provides the first non-trivial notion of sparsification for hypergraphs achievable with $O(n)$ hyperedges when $\epsilon$ and the rank $r$ of the hyperedges are constant. Finally, we provide a new construction of spectral hypergraph sparsifiers, according to the standard definition, with ${\rm poly}(\epsilon^{-1},r)\cdot n\log n$ hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used $\tilde O(n^3)$ hyperedges even for constant $r$ and $\epsilon$.Comment: 31 page

Beyond pairwise clustering

We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms