Mathematics in Independent Component Analysis

144 Chapter 9. Neurocomputing (in press), 2007
So both source separation models can be interpreted as matrix factorization
problems; in the temporal case restrictions such as diagonal autocorrelations
are determined by the second factor, in the spatial case by the first one.
In order to achieve a spatiotemporal model, we require these conditions from
both factors at the same time. In other words, instead of recovering a single
source data set which fulfills the source conditions spatiotemporally we try
to find two source matrices, a spatial and a temporal source matrix, and the
conditions are put onto the matrices separately. So the spatiotemporal BSS
model can be derived from equation (2) as the factorization problem
X = s S ⊤t S (3)
with spatial source matrix s S and temporal source matrix t S, which both
have (multidimensional) autocorrelations being as diagonal as possible. Diagonality
of the autocorrelations is invariant under scaling and permutation,
so the above model contains these indeterminacies — indeed the spatial and
temporal sources can interchange scaling (L) and permutation (P) matrices,
s S ⊤t S = (L −1 P −1s S) ⊤ (LP t S), and the model assumptions still hold. The spatiotemporal
BSS problem as defined in equation (3) has been implicitly proposed
in [9], equation (5), in combination with a dimension reduction scheme.
Here, we first operate on the general model and derive the cost function based
on autodecorrelation, and only later combine this with a dimension reduction
method.
5 Algorithmic spatiotemporal BSS
Stone et al. [9] first proposed the model from equation (3), where a joint energy
function is employed based on mutual entropy and Infomax. Apart from
the many parameters used in the algorithm, the involved gradient descent
optimization is susceptible to noise, local minima and inappropriate initializations,
so we propose a novel, more robust algebraic approach in the following.
It is based on the joint diagonalization of source conditions posed not only
temporally but also spatially at the same time.
5.1 Spatiotemporal BSS using joint diagonalization
Shifting to matrix notation, we interpret ¯ Rk(X) := ¯ Rk( t x(t)) as a symmetrized
temporal autocorrelation matrix, whereas ¯ Rk(X ⊤ ) := ¯ Rk( s x(r)) denotes
the corresponding spatial possibly multidimensional symmetrized autocorrelation
matrix. Here k indexes the one- or multidimensional lags τ. Ap-
5