251,660 research outputs found
Finding Dense Clusters via "Low Rank + Sparse" Decomposition
Finding "densely connected clusters" in a graph is in general an important and well studied problem in the literature. It has various applications in pattern recognition, social networking and data mining. Recently, Ames and Vavasis have suggested a novel method
for finding cliques in a graph by using convex optimization over the adjacency matrix of the graph. Also, there has been recent advances in decomposing a given matrix into its "low rank" and "sparse" components. In this paper, inspired by these results, we view "densely connected clusters" as imperfect cliques, where imperfections correspond missing edges, which are relatively sparse. We analyze the problem
in a probabilistic setting and aim to detect disjointly planted clusters. Our main result basically suggests that, one can find dense clusters in a graph, as long as the clusters are sufficiently large. We conclude by
discussing possible extensions and future research directions
Dense PGL-orbits in products of Grassmannians
In this paper, we find some necessary and sufficient conditions on the
dimension vector so that the diagonal
action of on has a dense orbit.
Consequently, we obtain some algorithms for finding dense and sparse dimension
vectors and classify dense dimension vectors with small length or size. We also
characterize the dense dimension vectors of the form .Comment: 21 page
Connected Choice and the Brouwer Fixed Point Theorem
We study the computational content of the Brouwer Fixed Point Theorem in the
Weihrauch lattice. Connected choice is the operation that finds a point in a
non-empty connected closed set given by negative information. One of our main
results is that for any fixed dimension the Brouwer Fixed Point Theorem of that
dimension is computably equivalent to connected choice of the Euclidean unit
cube of the same dimension. Another main result is that connected choice is
complete for dimension greater than or equal to two in the sense that it is
computably equivalent to Weak K\H{o}nig's Lemma. While we can present two
independent proofs for dimension three and upwards that are either based on a
simple geometric construction or a combinatorial argument, the proof for
dimension two is based on a more involved inverse limit construction. The
connected choice operation in dimension one is known to be equivalent to the
Intermediate Value Theorem; we prove that this problem is not idempotent in
contrast to the case of dimension two and upwards. We also prove that Lipschitz
continuity with Lipschitz constants strictly larger than one does not simplify
finding fixed points. Finally, we prove that finding a connectedness component
of a closed subset of the Euclidean unit cube of any dimension greater or equal
to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these
results, we introduce a representation of closed subsets of the unit cube by
trees of rational complexes.Comment: 36 page
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