1,711 research outputs found
Some upper and lower bounds on PSD-rank
Positive semidefinite rank (PSD-rank) is a relatively new quantity with
applications to combinatorial optimization and communication complexity. We
first study several basic properties of PSD-rank, and then develop new
techniques for showing lower bounds on the PSD-rank. All of these bounds are
based on viewing a positive semidefinite factorization of a matrix as a
quantum communication protocol. These lower bounds depend on the entries of the
matrix and not only on its support (the zero/nonzero pattern), overcoming a
limitation of some previous techniques. We compare these new lower bounds with
known bounds, and give examples where the new ones are better. As an
application we determine the PSD-rank of (approximations of) some common
matrices.Comment: 21 page
Estimating parameters of a multipartite loglinear graph model via the EM algorithm
We will amalgamate the Rash model (for rectangular binary tables) and the
newly introduced - models (for random undirected graphs) in the
framework of a semiparametric probabilistic graph model. Our purpose is to give
a partition of the vertices of an observed graph so that the generated
subgraphs and bipartite graphs obey these models, where their strongly
connected parameters give multiscale evaluation of the vertices at the same
time. In this way, a heterogeneous version of the stochastic block model is
built via mixtures of loglinear models and the parameters are estimated with a
special EM iteration. In the context of social networks, the clusters can be
identified with social groups and the parameters with attitudes of people of
one group towards people of the other, which attitudes depend on the cluster
memberships. The algorithm is applied to randomly generated and real-word data
On the geometric interpretation of the nonnegative rank
The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.nonnegative rank, restricted nonnegative rank, nested polytopes, computational complexity, computational geometry, extended formulations, linear Euclidean distance matrices.
Two Algorithms for Orthogonal Nonnegative Matrix Factorization with Application to Clustering
Approximate matrix factorization techniques with both nonnegativity and
orthogonality constraints, referred to as orthogonal nonnegative matrix
factorization (ONMF), have been recently introduced and shown to work
remarkably well for clustering tasks such as document classification. In this
paper, we introduce two new methods to solve ONMF. First, we show athematical
equivalence between ONMF and a weighted variant of spherical k-means, from
which we derive our first method, a simple EM-like algorithm. This also allows
us to determine when ONMF should be preferred to k-means and spherical k-means.
Our second method is based on an augmented Lagrangian approach. Standard ONMF
algorithms typically enforce nonnegativity for their iterates while trying to
achieve orthogonality at the limit (e.g., using a proper penalization term or a
suitably chosen search direction). Our method works the opposite way:
orthogonality is strictly imposed at each step while nonnegativity is
asymptotically obtained, using a quadratic penalty. Finally, we show that the
two proposed approaches compare favorably with standard ONMF algorithms on
synthetic, text and image data sets.Comment: 17 pages, 8 figures. New numerical experiments (document and
synthetic data sets
Nonnegative factorization and the maximum edge biclique problem
Nonnegative matrix factorization (NMF) is a data analysis technique based on the approximation of a nonnegative matrix with a product of two nonnegative factors, which allows compression and interpretation of nonnegative data. In this paper, we study the case of rank-one factorization and show that when the matrix to be factored is not required to be nonnegative, the corresponding problem (R1NF) becomes NP-hard. This sheds new light on the complexity of NMF since any algorithm for fixed-rank NMF must be able to solve at least implicitly such rank-one subproblems. Our proof relies on a reduction of the maximum edge biclique problem to R1NF. We also link stationary points of R1NF to feasible solutions of the biclique problem, which allows us to design a new type of biclique finding algorithm based on the application of a block-coordinate descent scheme to R1NF. We show that this algorithm, whose algorithmic complexity per iteration is proportional to the number of edges in the graph, is guaranteed to converge to a biclique and that it performs competitively with existing methods on random graphs and text mining datasets.nonnegative matrix factorization, rank-one factorization, maximum edge biclique problem, algorithmic complexity, biclique finding algorithm
An efficient and principled method for detecting communities in networks
A fundamental problem in the analysis of network data is the detection of
network communities, groups of densely interconnected nodes, which may be
overlapping or disjoint. Here we describe a method for finding overlapping
communities based on a principled statistical approach using generative network
models. We show how the method can be implemented using a fast, closed-form
expectation-maximization algorithm that allows us to analyze networks of
millions of nodes in reasonable running times. We test the method both on
real-world networks and on synthetic benchmarks and find that it gives results
competitive with previous methods. We also show that the same approach can be
used to extract nonoverlapping community divisions via a relaxation method, and
demonstrate that the algorithm is competitively fast and accurate for the
nonoverlapping problem.Comment: 14 pages, 5 figures, 1 tabl
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