Simple LU and QR Based Non-orthogonal Matrix Joint Diagonalization

Abstract

Abstract. A class of simple Jacobi-type algorithms for non-orthogonal matrix joint diagonalization based on the LU or QR factorization is introduced. By appropriate parametrization of the underlying manifolds, i.e. using triangular and orthogonal Jacobi matrices we replace a high dimensional minimization problem by a sequence of simple one dimensional minimization problems. In addition, a new scale-invariant cost function for non-orthogonal joint diagonalization is employed. These algorithms are step-size free. Numerical simulations demonstrate the efficiency of the methods.

Block Jacobi-type methods for non-orthogonal joint diagonalisation

“…More recently, to avoid these drawbacks of the functions (4) and (5), a new subspace fitting based cost function is developed in [2]. Following our analysis in the next section, however, we can show that this cost function does not share the features, which we will derive for both the off-norm function and the log-likelihood function.…”

“…In other words, the block Jacobi-type NoJD method developed in this work does not apply to it. Therefore, discussion and analysis on the subspace fitting function in [2] are omitted.…”

Nonnegative Joint Diagonalization by Congruence Based on LU Matrix Factorization

“…Consider a set of real symmetric matrices sharing the common structure: (1) where denotes an unknown transformation matrix, and is a set of unknown real diagonal matrices. is called a joint diagonalizer of .…”

“…Therefore we propose to impose a nonnegativity constraint on the matrix in the JDC problem. Generally, can be estimated either indirectly or directly [4]: 1) Indirect algorithms, such as JAD [3], FFDIAG [22], QDIAG [18], LUJ1D [1], FLEXJD [21], J-DI [14] and CVFFDIAG [19], estimate from the inverse of a transformation matrix , which minimizes the non-diagonal parts of using the following criterion:…”

“…These algorithms use additional constraint on , such as orthogonality [3] or unitdeterminant [1], in order to avoid the trivial solution . There also exist some other criterions.…”