24 research outputs found
On The Effect of Hyperedge Weights On Hypergraph Learning
Hypergraph is a powerful representation in several computer vision, machine
learning and pattern recognition problems. In the last decade, many researchers
have been keen to develop different hypergraph models. In contrast, no much
attention has been paid to the design of hyperedge weights. However, many
studies on pairwise graphs show that the choice of edge weight can
significantly influence the performances of such graph algorithms. We argue
that this also applies to hypegraphs. In this paper, we empirically discuss the
influence of hyperedge weight on hypegraph learning via proposing three novel
hyperedge weights from the perspectives of geometry, multivariate statistical
analysis and linear regression. Extensive experiments on ORL, COIL20, JAFFE,
Sheffield, Scene15 and Caltech256 databases verify our hypothesis. Similar to
graph learning, several representative hyperedge weighting schemes can be
concluded by our experimental studies. Moreover, the experiments also
demonstrate that the combinations of such weighting schemes and conventional
hypergraph models can get very promising classification and clustering
performances in comparison with some recent state-of-the-art algorithms
Hypergraph Learning with Line Expansion
Previous hypergraph expansions are solely carried out on either vertex level
or hyperedge level, thereby missing the symmetric nature of data co-occurrence,
and resulting in information loss. To address the problem, this paper treats
vertices and hyperedges equally and proposes a new hypergraph formulation named
the \emph{line expansion (LE)} for hypergraphs learning. The new expansion
bijectively induces a homogeneous structure from the hypergraph by treating
vertex-hyperedge pairs as "line nodes". By reducing the hypergraph to a simple
graph, the proposed \emph{line expansion} makes existing graph learning
algorithms compatible with the higher-order structure and has been proven as a
unifying framework for various hypergraph expansions. We evaluate the proposed
line expansion on five hypergraph datasets, the results show that our method
beats SOTA baselines by a significant margin
On the spectrum of hypergraphs
Here we study the spectral properties of an underlying weighted graph of a
non-uniform hypergraph by introducing different connectivity matrices, such as
adjacency, Laplacian and normalized Laplacian matrices. We show that different
structural properties of a hypergrpah, can be well studied using spectral
properties of these matrices. Connectivity of a hypergraph is also investigated
by the eigenvalues of these operators. Spectral radii of the same are bounded
by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by
the eigenvalues of its connectivity matrices. We characterize different
properties of a regular hypergraph characterized by the spectrum. Strong
(vertex) chromatic number of a hypergraph is bounded by the eigenvalues.
Cheeger constant on a hypergraph is defined and we show that it can be bounded
by the smallest nontrivial eigenvalues of Laplacian matrix and normalized
Laplacian matrix, respectively, of a connected hypergraph. We also show an
approach to study random walk on a (non-uniform) hypergraph that can be
performed by analyzing the spectrum of transition probability operator which is
defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two
different ways. We show that if the Laplace operator, , on a hypergraph
satisfies a curvature-dimension type inequality
with and then any non-zero eigenvalue of can be bounded below by . Eigenvalues of a normalized Laplacian operator defined on a connected
hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph
Hypergraph -Laplacian: A Differential Geometry View
The graph Laplacian plays key roles in information processing of relational
data, and has analogies with the Laplacian in differential geometry. In this
paper, we generalize the analogy between graph Laplacian and differential
geometry to the hypergraph setting, and propose a novel hypergraph
-Laplacian. Unlike the existing two-node graph Laplacians, this
generalization makes it possible to analyze hypergraphs, where the edges are
allowed to connect any number of nodes. Moreover, we propose a semi-supervised
learning method based on the proposed hypergraph -Laplacian, and formalize
them as the analogue to the Dirichlet problem, which often appears in physics.
We further explore theoretical connections to normalized hypergraph cut on a
hypergraph, and propose normalized cut corresponding to hypergraph
-Laplacian. The proposed -Laplacian is shown to outperform standard
hypergraph Laplacians in the experiment on a hypergraph semi-supervised
learning and normalized cut setting.Comment: Extended version of our AAAI-18 pape
Balanced Coarsening for Multilevel Hypergraph Partitioning via Wasserstein Discrepancy
We propose a balanced coarsening scheme for multilevel hypergraph
partitioning. In addition, an initial partitioning algorithm is designed to
improve the quality of k-way hypergraph partitioning. By assigning vertex
weights through the LPT algorithm, we generate a prior hypergraph under a
relaxed balance constraint. With the prior hypergraph, we have defined the
Wasserstein discrepancy to coordinate the optimal transport of coarsening
process. And the optimal transport matrix is solved by Sinkhorn algorithm. Our
coarsening scheme fully takes into account the minimization of connectivity
metric (objective function). For the initial partitioning stage, we define a
normalized cut function induced by Fiedler vector, which is theoretically
proved to be a concave function. Thereby, a three-point algorithm is designed
to find the best cut under the balance constraint
View-aligned hypergraph learning for Alzheimer’s disease diagnosis with incomplete multi-modality data
AbstractEffectively utilizing incomplete multi-modality data for the diagnosis of Alzheimer's disease (AD) and its prodrome (i.e., mild cognitive impairment, MCI) remains an active area of research. Several multi-view learning methods have been recently developed for AD/MCI diagnosis by using incomplete multi-modality data, with each view corresponding to a specific modality or a combination of several modalities. However, existing methods usually ignore the underlying coherence among views, which may lead to sub-optimal learning performance. In this paper, we propose a view-aligned hypergraph learning (VAHL) method to explicitly model the coherence among views. Specifically, we first divide the original data into several views based on the availability of different modalities and then construct a hypergraph in each view space based on sparse representation. A view-aligned hypergraph classification (VAHC) model is then proposed, by using a view-aligned regularizer to capture coherence among views. We further assemble the class probability scores generated from VAHC, via a multi-view label fusion method for making a final classification decision. We evaluate our method on the baseline ADNI-1 database with 807 subjects and three modalities (i.e., MRI, PET, and CSF). Experimental results demonstrate that our method outperforms state-of-the-art methods that use incomplete multi-modality data for AD/MCI diagnosis