6,163 research outputs found
On Weighted Low-Rank Approximation
Our main interest is the low-rank approximation of a matrix in R^m.n under a
weighted Frobenius norm. This norm associates a weight to each of the (m x n)
matrix entries. We conjecture that the number of approximations is at most
min(m, n).
We also investigate how the approximations depend on the weight-values.Comment: 13 page
Rank-1 Tensor Approximation Methods and Application to Deflation
Because of the attractiveness of the canonical polyadic (CP) tensor
decomposition in various applications, several algorithms have been designed to
compute it, but efficient ones are still lacking. Iterative deflation
algorithms based on successive rank-1 approximations can be used to perform
this task, since the latter are rather easy to compute. We first present an
algebraic rank-1 approximation method that performs better than the standard
higher-order singular value decomposition (HOSVD) for three-way tensors.
Second, we propose a new iterative rank-1 approximation algorithm that improves
any other rank-1 approximation method. Third, we describe a probabilistic
framework allowing to study the convergence of deflation CP decomposition
(DCPD) algorithms based on successive rank-1 approximations. A set of computer
experiments then validates theoretical results and demonstrates the efficiency
of DCPD algorithms compared to other ones
Rank-preserving geometric means of positive semi-definite matrices
The generalization of the geometric mean of positive scalars to positive
definite matrices has attracted considerable attention since the seminal work
of Ando. The paper generalizes this framework of matrix means by proposing the
definition of a rank-preserving mean for two or an arbitrary number of positive
semi-definite matrices of fixed rank. The proposed mean is shown to be
geometric in that it satisfies all the expected properties of a rank-preserving
geometric mean. The work is motivated by operations on low-rank approximations
of positive definite matrices in high-dimensional spaces.Comment: To appear in Linear Algebra and its Application
An Interpretation of the Dual Problem of the THREE-like Approaches
Spectral estimation can be preformed using the so called THREE-like approach.
Such method leads to a convex optimization problem whose solution is
characterized through its dual problem. In this paper, we show that the dual
problem can be seen as a new parametric spectral estimation problem. This
interpretation implies that the THREE-like solution is optimal in terms of
closeness to the correlogram over a certain parametric class of spectral
densities, enriching in this way its meaningfulness
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