10,761 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
Learning Hypergraph-regularized Attribute Predictors
We present a novel attribute learning framework named Hypergraph-based
Attribute Predictor (HAP). In HAP, a hypergraph is leveraged to depict the
attribute relations in the data. Then the attribute prediction problem is
casted as a regularized hypergraph cut problem in which HAP jointly learns a
collection of attribute projections from the feature space to a hypergraph
embedding space aligned with the attribute space. The learned projections
directly act as attribute classifiers (linear and kernelized). This formulation
leads to a very efficient approach. By considering our model as a multi-graph
cut task, our framework can flexibly incorporate other available information,
in particular class label. We apply our approach to attribute prediction,
Zero-shot and -shot learning tasks. The results on AWA, USAA and CUB
databases demonstrate the value of our methods in comparison with the
state-of-the-art approaches.Comment: This is an attribute learning paper accepted by CVPR 201
Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space
The rigidity of the Positive Mass Theorem states that the only complete
asymptotically flat manifold of nonnegative scalar curvature and zero mass is
Euclidean space. We study the stability of this statement for spaces that can
be realized as graphical hypersurfaces in Euclidean space. We prove (under
certain technical hypotheses) that if a sequence of complete asymptotically
flat graphs of nonnegative scalar curvature has mass approaching zero, then the
sequence must converge to Euclidean space in the pointed intrinsic flat sense.
The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness
theorem for sequences of metric spaces with uniform Lipschitz bounds on their
metrics.Comment: 31 pages, 2 figures, v2: to appear in Crelle's Journal, many minor
changes, one new exampl
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