518 research outputs found
Simplified Energy Landscape for Modularity Using Total Variation
Networks capture pairwise interactions between entities and are frequently
used in applications such as social networks, food networks, and protein
interaction networks, to name a few. Communities, cohesive groups of nodes,
often form in these applications, and identifying them gives insight into the
overall organization of the network. One common quality function used to
identify community structure is modularity. In Hu et al. [SIAM J. App. Math.,
73(6), 2013], it was shown that modularity optimization is equivalent to
minimizing a particular nonconvex total variation (TV) based functional over a
discrete domain. They solve this problem, assuming the number of communities is
known, using a Merriman, Bence, Osher (MBO) scheme.
We show that modularity optimization is equivalent to minimizing a convex
TV-based functional over a discrete domain, again, assuming the number of
communities is known. Furthermore, we show that modularity has no convex
relaxation satisfying certain natural conditions. We therefore, find a
manageable non-convex approximation using a Ginzburg Landau functional, which
provably converges to the correct energy in the limit of a certain parameter.
We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et
al. and which is 7 times faster at solving the associated diffusion equation
due to the fact that the underlying discretization is unconditionally stable.
Our numerical tests include a hyperspectral video whose associated graph has
2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper
of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat
Disconnected Skeleton: Shape at its Absolute Scale
We present a new skeletal representation along with a matching framework to
address the deformable shape recognition problem. The disconnectedness arises
as a result of excessive regularization that we use to describe a shape at an
attainably coarse scale. Our motivation is to rely on the stable properties of
the shape instead of inaccurately measured secondary details. The new
representation does not suffer from the common instability problems of
traditional connected skeletons, and the matching process gives quite
successful results on a diverse database of 2D shapes. An important difference
of our approach from the conventional use of the skeleton is that we replace
the local coordinate frame with a global Euclidean frame supported by
additional mechanisms to handle articulations and local boundary deformations.
As a result, we can produce descriptions that are sensitive to any combination
of changes in scale, position, orientation and articulation, as well as
invariant ones.Comment: The work excluding {\S}V and {\S}VI has first appeared in 2005 ICCV:
Aslan, C., Tari, S.: An Axis-Based Representation for Recognition. In
ICCV(2005) 1339- 1346.; Aslan, C., : Disconnected Skeletons for Shape
Recognition. Masters thesis, Department of Computer Engineering, Middle East
Technical University, May 200
Probabilistic Properties of Highly Connected Random Geometric Graphs
In this paper we study the probabilistic properties of reliable networks of minimal total edge lengths. We study reliability in terms of k-edge-connectivity in graphs in d-dimensional space. We show this problem fits into Yukich’s framework for Euclidean functionals for arbitrary k, dimension d and distant-power gradient p, with p < d. With this framework several theorems on the convergence of optimal solutions follow. We apply Yukich’s framework for functionals so that we can use partitioning algorithms that rapidly compute near-optimal solutions on typical examples. These results are then extended to optimal k-edge-connected power assignment graphs, where we assign power to vertices and charge per vertex. The network can be modelled as a wireless network
Parsimonious Time Series Clustering
We introduce a parsimonious model-based framework for clustering time course
data. In these applications the computational burden becomes often an issue due
to the number of available observations. The measured time series can also be
very noisy and sparse and a suitable model describing them can be hard to
define. We propose to model the observed measurements by using P-spline
smoothers and to cluster the functional objects as summarized by the optimal
spline coefficients. In principle, this idea can be adopted within all the most
common clustering frameworks. In this work we discuss applications based on a
k-means algorithm. We evaluate the accuracy and the efficiency of our proposal
by simulations and by dealing with drosophila melanogaster gene expression
data
Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using -Means Type Optimization
We address the task of estimating multiple trajectories from unlabeled data.
This problem arises in many settings, one could think of the construction of
maps of transport networks from passive observation of travellers, or the
reconstruction of the behaviour of uncooperative vehicles from external
observations, for example. There are two coupled problems. The first is a data
association problem: how to map data points onto individual trajectories. The
second is, given a solution to the data association problem, to estimate those
trajectories. We construct estimators as a solution to a regularized
variational problem (to which approximate solutions can be obtained via the
simple, efficient and widespread -means method) and show that, as the number
of data points, , increases, these estimators exhibit stable behaviour. More
precisely, we show that they converge in an appropriate Sobolev space in
probability and with rate
Colour, texture, and motion in level set based segmentation and tracking
This paper introduces an approach for the extraction and combination of different cues in a level set based image segmentation framework. Apart from the image grey value or colour, we suggest to add its spatial and temporal variations, which may provide important further characteristics. It often turns out that the combination of colour, texture, and motion permits to distinguish object regions that cannot be separated by one cue alone. We propose a two-step approach. In the first stage, the input features are extracted and enhanced by applying coupled nonlinear diffusion. This ensures coherence between the channels and deals with outliers. We use a nonlinear diffusion technique, closely related to total variation flow, but being strictly edge enhancing. The resulting features are then employed for a vector-valued front propagation based on level sets and statistical region models that approximate the distributions of each feature. The application of this approach to two-phase segmentation is followed by an extension to the tracking of multiple objects in image sequences
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