452 research outputs found
On The Isoperimetric Spectrum of Graphs and Its Approximations
In this paper we consider higher isoperimetric numbers of a (finite directed)
graph. In this regard we focus on the th mean isoperimetric constant of a
directed graph as the minimum of the mean outgoing normalized flows from a
given set of disjoint subsets of the vertex set of the graph. We show that
the second mean isoperimetric constant in this general setting, coincides with
(the mean version of) the classical Cheeger constant of the graph, while for
the rest of the spectrum we show that there is a fundamental difference between
the th isoperimetric constant and the number obtained by taking the minimum
over all -partitions. In this direction, we show that our definition is the
correct one in the sense that it satisfies a Federer-Fleming-type theorem, and
we also define and present examples for the concept of a supergeometric graph
as a graph whose mean isoperimetric constants are attained on partitions at all
levels. Moreover, considering the -completeness of the isoperimetric
problem on graphs, we address ourselves to the approximation problem where we
prove general spectral inequalities that give rise to a general Cheeger-type
inequality as well. On the other hand, we also consider some algorithmic
aspects of the problem where we show connections to orthogonal representations
of graphs and following J.~Malik and J.~Shi () we study the close
relationships to the well-known -means algorithm and normalized cuts method
Efficient Mining of Heterogeneous Star-Structured Data
Many of the real world clustering problems arising in data mining applications are heterogeneous in nature. Heterogeneous co-clustering involves simultaneous clustering of objects of two or more data types. While pairwise co-clustering of two data types has been well studied in the literature, research on high-order heterogeneous co-clustering is still limited. In this paper, we propose a graph theoretical framework for addressing star- structured co-clustering problems in which a central data type is connected to all the other data types. Partitioning this graph leads to co-clustering of all the data types under the constraints of the star-structure. Although, graph partitioning approach has been adopted before to address star-structured heterogeneous complex problems, the main contribution of this work lies in an e cient algorithm that we propose for partitioning the star-structured graph. Computationally, our algorithm is very quick as it requires a simple solution to a sparse system of overdetermined linear equations. Theoretical analysis and extensive exper- iments performed on toy and real datasets demonstrate the quality, e ciency and stability of the proposed algorithm
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Convex Geometry and its Applications (hybrid meeting)
The geometry of convex domains in Euclidean space plays a central role
in several branches of mathematics: functional and harmonic analysis, the
theory of PDE, linear programming and, increasingly, in the study of
algorithms in computer science.
The purpose
of this meeting was to bring together researchers from the analytic, geometric and probabilistic
groups who have contributed to these developments
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
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