455 research outputs found
Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem
In this paper, we introduce properly-invariant diagonality measures of
Hermitian positive-definite matrices. These diagonality measures are defined as
distances or divergences between a given positive-definite matrix and its
diagonal part. We then give closed-form expressions of these diagonality
measures and discuss their invariance properties. The diagonality measure based
on the log-determinant -divergence is general enough as it includes a
diagonality criterion used by the signal processing community as a special
case. These diagonality measures are then used to formulate minimization
problems for finding the approximate joint diagonalizer of a given set of
Hermitian positive-definite matrices. Numerical computations based on a
modified Newton method are presented and commented
Non-orthogonal joint block diagonalization based on the LU or QR factorizations for convolutive blind source separation
This article addresses the problem of blind source separation, in which the source signals are most often of the convolutive mixtures, and moreover, the source signals cannot satisfy independent identical distribution generally. One kind of prevailing and representative approaches for overcoming these difficulties is joint block diagonalization (JBD) method. To improve present JBD methods, we present a class of simple Jacobi-type JBD algorithms based on the LU or QR factorizations. Using Jacobi-type matrices we can replace high dimensional minimization problems with a sequence of simple one-dimensional problems. The novel methods are more general i.e. the orthogonal, positive definite or symmetric matrices and a preliminary whitening stage is no more compulsorily required, and further, the convergence is also guaranteed. The performance of the proposed algorithms, compared with the existing state-of-the-art JBD algorithms, is evaluated with computer simulations and vibration experimental. The results of numerical examples demonstrate that the robustness and effectiveness of the two novel algorithms provide a significant improvement i.e., yield less convergence time, higher precision of convergence, better success rate of block diagonalization. And the proposed algorithms are effective in separating the vibration signals of convolutive mixtures
Randomized Joint Diagonalization of Symmetric Matrices
Given a family of nearly commuting symmetric matrices, we consider the task
of computing an orthogonal matrix that nearly diagonalizes every matrix in the
family. In this paper, we propose and analyze randomized joint diagonalization
(RJD) for performing this task. RJD applies a standard eigenvalue solver to
random linear combinations of the matrices. Unlike existing optimization-based
methods, RJD is simple to implement and leverages existing high-quality linear
algebra software packages. Our main novel contribution is to prove robust
recovery: Given a family that is -near to a commuting family, RJD
jointly diagonalizes this family, with high probability, up to an error of norm
O(). No other existing method is known to enjoy such a universal
robust recovery guarantee. We also discuss how the algorithm can be further
improved by deflation techniques and demonstrate its state-of-the-art
performance by numerical experiments with synthetic and real-world data
Newton-Type Methods For Simultaneous Matrix Diagonalization
This paper proposes a Newton-type method to solve numerically the
eigenproblem of several diagonalizable matrices, which pairwise commute. A
classical result states that these matrices are simultaneously diagonalizable.
From a suitable system of equations associated to this problem, we construct a
sequence that converges quadratically towards the solution. This construction
is not based on the resolution of a linear system as is the case in the
classical Newton method. Moreover, we provide a theoretical analysis of this
construction and exhibit a condition to get a quadratic convergence. We also
propose numerical experiments, which illustrate the theoretical results.Comment: Calcolo, Springer Verlag, 202
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