279 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Sublinear algorithms for local graph centrality estimation
We study the complexity of local graph centrality estimation, with the goal
of approximating the centrality score of a given target node while exploring
only a sublinear number of nodes/arcs of the graph and performing a sublinear
number of elementary operations. We develop a technique, that we apply to the
PageRank and Heat Kernel centralities, for building a low-variance score
estimator through a local exploration of the graph. We obtain an algorithm
that, given any node in any graph of arcs, with probability
computes a multiplicative -approximation of its score by
examining only nodes/arcs, where and are respectively the maximum and
average outdegree of the graph (omitting for readability
and
factors). A similar bound holds for computational complexity. We also prove a
lower bound of for both query complexity and computational complexity. Moreover,
our technique yields a query complexity algorithm for the
graph access model of [Brautbar et al., 2010], widely used in social network
mining; we show this algorithm is optimal up to a sublogarithmic factor. These
are the first algorithms yielding worst-case sublinear bounds for general
directed graphs and any choice of the target node.Comment: 29 pages, 1 figur
A discrete graph Laplacian for signal processing
In this thesis we exploit diffusion processes on graphs to effect two fundamental problems of image processing: denoising and segmentation. We treat these two low-level vision problems on the pixel-wise level under a unified framework: a graph embedding. Using this framework opens us up to the possibilities of exploiting recently introduced algorithms from the semi-supervised machine learning literature.
We contribute two novel edge-preserving smoothing algorithms to the literature. Furthermore we apply these edge-preserving smoothing algorithms to some computational photography tasks. Many recent computational photography tasks require the decomposition of an image into a smooth base layer containing large scale intensity variations and a residual layer capturing fine details. Edge-preserving smoothing is the main computational mechanism in producing these multi-scale image representations. We, in effect, introduce a new approach to edge-preserving multi-scale image decompositions. Where as prior approaches such as the Bilateral filter and weighted-least squares methods require multiple parameters to tune the response of the filters our method only requires one. This parameter can be interpreted as a scale parameter. We demonstrate the utility of our approach by applying the method to computational photography tasks that utilise multi-scale image decompositions.
With minimal modification to these edge-preserving smoothing algorithms we show that we can extend them to produce interactive image segmentation. As a result the operations of segmentation and denoising are conducted under a unified framework. Moreover we discuss how our method is related to region based active contours. We benchmark our proposed interactive segmentation algorithms against those based upon energy-minimisation, specifically graph-cut methods. We demonstrate that we achieve competitive performance
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