1,552 research outputs found
Modularity functions maximization with nonnegative relaxation facilitates community detection in networks
We show here that the problem of maximizing a family of quantitative
functions, encompassing both the modularity (Q-measure) and modularity density
(D-measure), for community detection can be uniformly understood as a
combinatoric optimization involving the trace of a matrix called modularity
Laplacian. Instead of using traditional spectral relaxation, we apply
additional nonnegative constraint into this graph clustering problem and design
efficient algorithms to optimize the new objective. With the explicit
nonnegative constraint, our solutions are very close to the ideal community
indicator matrix and can directly assign nodes into communities. The
near-orthogonal columns of the solution can be reformulated as the posterior
probability of corresponding node belonging to each community. Therefore, the
proposed method can be exploited to identify the fuzzy or overlapping
communities and thus facilitates the understanding of the intrinsic structure
of networks. Experimental results show that our new algorithm consistently,
sometimes significantly, outperforms the traditional spectral relaxation
approaches
Heavy-tailed Independent Component Analysis
Independent component analysis (ICA) is the problem of efficiently recovering
a matrix from i.i.d. observations of
where is a random vector with mutually independent
coordinates. This problem has been intensively studied, but all existing
efficient algorithms with provable guarantees require that the coordinates
have finite fourth moments. We consider the heavy-tailed ICA problem
where we do not make this assumption, about the second moment. This problem
also has received considerable attention in the applied literature. In the
present work, we first give a provably efficient algorithm that works under the
assumption that for constant , each has finite
-moment, thus substantially weakening the moment requirement
condition for the ICA problem to be solvable. We then give an algorithm that
works under the assumption that matrix has orthogonal columns but requires
no moment assumptions. Our techniques draw ideas from convex geometry and
exploit standard properties of the multivariate spherical Gaussian distribution
in a novel way.Comment: 30 page
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