78 research outputs found
About the maximal rank of 3-tensors over the real and the complex number field
High dimensional array data, tensor data, is becoming important in recent
days. Then maximal rank of tensors is important in theory and applications. In
this paper we consider the maximal rank of 3 tensors. It can be attacked from
various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and
Atkinson-Lloyd(1980). They treated the problem in the complex field, and we
will present various bounds over the real field by proving several lemmas and
propositions, which is real counterparts of their results.Comment: 13 pages, no figure v2: correction and improvemen
The Iterative Signature Algorithm for the analysis of large scale gene expression data
We present a new approach for the analysis of genome-wide expression data.
Our method is designed to overcome the limitations of traditional techniques,
when applied to large-scale data. Rather than alloting each gene to a single
cluster, we assign both genes and conditions to context-dependent and
potentially overlapping transcription modules. We provide a rigorous definition
of a transcription module as the object to be retrieved from the expression
data. An efficient algorithm, that searches for the modules encoded in the data
by iteratively refining sets of genes and conditions until they match this
definition, is established. Each iteration involves a linear map, induced by
the normalized expression matrix, followed by the application of a threshold
function. We argue that our method is in fact a generalization of Singular
Value Decomposition, which corresponds to the special case where no threshold
is applied. We show analytically that for noisy expression data our approach
leads to better classification due to the implementation of the threshold. This
result is confirmed by numerical analyses based on in-silico expression data.
We discuss briefly results obtained by applying our algorithm to expression
data from the yeast S. cerevisiae.Comment: Latex, 36 pages, 8 figure
Branch and bound based coordinate search filter algorithm for nonsmooth nonconvex mixed-integer nonlinear programming problems
Publicado em "Computational science and its applications – ICCSA 2014...", ISBN 978-3-319-09128-0. Series "Lecture notes in computer science", ISSN 0302-9743, vol. 8580.A mixed-integer nonlinear programming problem (MINLP) is a problem with continuous and integer variables and at least, one nonlinear function. This kind of problem appears in a wide range of real applications and is very difficult to solve. The difficulties are due to the nonlinearities of the functions in the problem and the integrality restrictions on some variables. When they are nonconvex then they are the most difficult to solve above all. We present a methodology to solve nonsmooth nonconvex MINLP problems based on a branch and bound paradigm and a stochastic strategy. To solve the relaxed subproblems at each node of the branch and bound tree search, an algorithm based on a multistart strategy with a coordinate search filter methodology is implemented. The produced numerical results show the robustness of the proposed methodology.This work has been supported by FCT (Fundação para a Ciência e aTecnologia) in the scope of the projects: PEst-OE/MAT/UI0013/2014 and PEst-OE/EEI/UI0319/2014
Tensor completion in hierarchical tensor representations
Compressed sensing extends from the recovery of sparse vectors from
undersampled measurements via efficient algorithms to the recovery of matrices
of low rank from incomplete information. Here we consider a further extension
to the reconstruction of tensors of low multi-linear rank in recently
introduced hierarchical tensor formats from a small number of measurements.
Hierarchical tensors are a flexible generalization of the well-known Tucker
representation, which have the advantage that the number of degrees of freedom
of a low rank tensor does not scale exponentially with the order of the tensor.
While corresponding tensor decompositions can be computed efficiently via
successive applications of (matrix) singular value decompositions, some
important properties of the singular value decomposition do not extend from the
matrix to the tensor case. This results in major computational and theoretical
difficulties in designing and analyzing algorithms for low rank tensor
recovery. For instance, a canonical analogue of the tensor nuclear norm is
NP-hard to compute in general, which is in stark contrast to the matrix case.
In this book chapter we consider versions of iterative hard thresholding
schemes adapted to hierarchical tensor formats. A variant builds on methods
from Riemannian optimization and uses a retraction mapping from the tangent
space of the manifold of low rank tensors back to this manifold. We provide
first partial convergence results based on a tensor version of the restricted
isometry property (TRIP) of the measurement map. Moreover, an estimate of the
number of measurements is provided that ensures the TRIP of a given tensor rank
with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its
Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
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