Dependent Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 142

to be extracted (i.e. n m ).

The signals x ( t ) are observed, and the goal is to recover A and s ( t ).
Having found A , s ( t ) can be estimated by A † x ( t ), which is optimal in
the maximum-likelihood sense. Here † denotes the pseudo inverse of A ,
which equals the inverse in the case of m = n . Thus the BSS task reduces
to the estimation of the mixing matrix A , and hence, the additive noise
n is often neglected (after whitening). Note that in the following we will
assume that all signals are real-valued. Extensions to the complex case
are straightforward.

≤

Approximate joint diagonalization

Many BSS algorithms employ joint diagonalization (JD) techniques on
some source condition matrices to identify the mixing matrix. Given a set
of symmetric matrices

, JD implies minimizing the
squared sum of the off-diagonal elements of A C i A , that is minimizing

C := { C 1 ,..., C K }

K

f ( A ):=

A C i A

diag( A C i A ) F − (5.1)

i=1

with respect to the orthogonal matrix A , where diag( C ) produces a
matrix, where all off-diagonal elements of C have been set to zero,
and where F := tr( CC ) denotes the squared Frobenius norm. A

global minimum A of f is called a joint diagonalizer of

C

C . Such a joint

C

diagonalizer exists if and only if all elements of commute.

Algorithms for performing joint diagonalization include gradient de-
scent on f ( A ), Jacobi-like iterative construction of A by Givens rotation
in two coordinates [42], an extension minimizing a logarithmic version
of equation (5.1) [202], an alternating optimization scheme switching
between column and diagonal optimization [292], and, more recently, a
linear least-squares algorithm for diagonalization [297]. The latter three
algorithms can also search for non-orthogonal matrices A .Notethatin
practice, minimization of the off-sums yields only an approximate joint
diagonalizer —in the case of finite samples, the source condition matrices
are estimates. Hence they only approximately share the same eigenstruc-
ture and do not fully commutate, so f ( A ) from equation (5.1) cannot
be rendered zero precisely but only approximately.