371 research outputs found
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
Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering
We study p-Laplacians and spectral clustering for a recently proposed
hypergraph model that incorporates edge-dependent vertex weights (EDVW). These
weights can reflect different importance of vertices within a hyperedge, thus
conferring the hypergraph model higher expressivity and flexibility. By
constructing submodular EDVW-based splitting functions, we convert hypergraphs
with EDVW into submodular hypergraphs for which the spectral theory is better
developed. In this way, existing concepts and theorems such as p-Laplacians and
Cheeger inequalities proposed under the submodular hypergraph setting can be
directly extended to hypergraphs with EDVW. For submodular hypergraphs with
EDVW-based splitting functions, we propose an efficient algorithm to compute
the eigenvector associated with the second smallest eigenvalue of the
hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the
vertices, achieving higher clustering accuracy than traditional spectral
clustering based on the 2-Laplacian. More broadly, the proposed algorithm works
for all submodular hypergraphs that are graph reducible. Numerical experiments
using real-world data demonstrate the effectiveness of combining spectral
clustering based on the 1-Laplacian and EDVW
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
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