jedi {jointDiag}

Approximate non-orthogonal joint diagonalization of a set of square real-valued matrices

Description

This function performs a Joint Approximate Diagonalization of a set of square and real-valued matrices (not necessarily symmetric). The algorithm seeks the inverse of the joint diagonalizer (the mixing matrix in terms of source separation).

The algorithm uses Givens and hyperbolic rotations to find the inverse of a non-orthogonal joint diagonalizer. It is an extension of the JADE method (orthogonal joint diagonalization).

Usage

jedi(M, A0 = NULL, eps = .Machine$double.eps, itermax = 200, 
		keepTrace = FALSE)

Arguments

M

DOUBLE ARRAY (KxKxN). Three-dimensional array with dimensions KxKxN representing the set of square and real-valued matrices to be jointly diagonalized. N is the number of matrices. Matrices are KxK square matrices.

A0

DOUBLE MATRIX (KxK). The initial guess of the inverse of a joint diagonalizer. If NULL, an initial guess is automatically generated by the algorithm.

eps

DOUBLE. The algorithm stops when the criterium difference between two iterations is less than eps.

itermax

INTEGER. Alternatively, the algorithm stops when itermax sweeps have been performed without reaching convergence. If the maximum number of iteration is performed, a warning appears.

keepTrace

BOOLEAN. Do we want to keep the successive estimations of the joint diagonalizer.

Details

Given a set M_i of N K \times K square and real-valued matrices, the algorithm is looking for a matrix A such that \forall i \in [1,N], A^{-1} C_i A^{-T} is as close as possible of a diagonal matrix.

Value

A

Estimation of the Joint Diagonalizer.

criter

Successive estimates of the cost function across sweeps.

A_trace

Array of the successive estimates of A across iterations.

Warning

This algorithm based on a combination of givens and hyperbolic rotations is covered by a patent (see A. Souloumiac, CEA Saclay).

Author(s)

Cedric Gouy-Pailler ([e-mail]), with help from Antoine Souloumiac.

References

Souloumiac, A.; Non-Orthogonal Joint Diagonalization by Combining Givens and Hyperbolic Rotations; IEEE Trans. Signal Process., 2009

Examples

# generating diagonal matrices
D <- replicate(30, diag(rchisq(df=1,n=10)), simplify=FALSE)
# Mixing and demixing matrices
B <- matrix(rnorm(100),10,10)
A <- solve(B)
C <- array(NA,dim=c(10,10,30))
for (i in 1:30) C[,,i] <- A %*% D[[i]] %*% t(A)
A_est <- jedi(C)$A
# A_est should be an approximate of A
B %*% A_est
# close to a permutation matrix (with random scales)