138 research outputs found
Numerical Optimization for Symmetric Tensor Decomposition
We consider the problem of decomposing a real-valued symmetric tensor as the
sum of outer products of real-valued vectors. Algebraic methods exist for
computing complex-valued decompositions of symmetric tensors, but here we focus
on real-valued decompositions, both unconstrained and nonnegative, for problems
with low-rank structure. We discuss when solutions exist and how to formulate
the mathematical program. Numerical results show the properties of the proposed
formulations (including one that ignores symmetry) on a set of test problems
and illustrate that these straightforward formulations can be effective even
though the problem is nonconvex
Shifted Power Method for Computing Tensor Eigenpairs
Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has
been motivated by applications in blind source separation, magnetic resonance
imaging, molecular conformation, and more. In this paper, we consider methods
for computing real symmetric-tensor eigenpairs of the form Ax^{m-1} = \lambda x
subject to ||x||=1, which is closely related to optimal rank-1 approximation of
a symmetric tensor. Our contribution is a shifted symmetric higher-order power
method (SS-HOPM), which we show is guaranteed to converge to a tensor
eigenpair. SS-HOPM can be viewed as a generalization of the power iteration
method for matrices or of the symmetric higher-order power method.
Additionally, using fixed point analysis, we can characterize exactly which
eigenpairs can and cannot be found by the method. Numerical examples are
presented, including examples from an extension of the method to finding
complex eigenpairs
Triadic Measures on Graphs: The Power of Wedge Sampling
Graphs are used to model interactions in a variety of contexts, and there is
a growing need to quickly assess the structure of a graph. Some of the most
useful graph metrics, especially those measuring social cohesion, are based on
triangles. Despite the importance of these triadic measures, associated
algorithms can be extremely expensive. We propose a new method based on wedge
sampling. This versatile technique allows for the fast and accurate
approximation of all current variants of clustering coefficients and enables
rapid uniform sampling of the triangles of a graph. Our methods come with
provable and practical time-approximation tradeoffs for all computations. We
provide extensive results that show our methods are orders of magnitude faster
than the state-of-the-art, while providing nearly the accuracy of full
enumeration. Our results will enable more wide-scale adoption of triadic
measures for analysis of extremely large graphs, as demonstrated on several
real-world examples
Using Triangles to Improve Community Detection in Directed Networks
In a graph, a community may be loosely defined as a group of nodes that are
more closely connected to one another than to the rest of the graph. While
there are a variety of metrics that can be used to specify the quality of a
given community, one common theme is that flows tend to stay within
communities. Hence, we expect cycles to play an important role in community
detection. For undirected graphs, the importance of triangles -- an undirected
3-cycle -- has been known for a long time and can be used to improve community
detection. In directed graphs, the situation is more nuanced. The smallest
cycle is simply two nodes with a reciprocal connection, and using information
about reciprocation has proven to improve community detection. Our new idea is
based on the four types of directed triangles that contain cycles. To identify
communities in directed networks, then, we propose an undirected edge-weighting
scheme based on the type of the directed triangles in which edges are involved.
We also propose a new metric on quality of the communities that is based on the
number of 3-cycles that are split across communities. To demonstrate the impact
of our new weighting, we use the standard METIS graph partitioning tool to
determine communities and show experimentally that the resulting communities
result in fewer 3-cycles being cut. The magnitude of the effect varies between
a 10 and 50% reduction, and we also find evidence that this weighting scheme
improves a task where plausible ground-truth communities are known.Comment: 10 pages, 3 figure
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