Summary

ICA can be summarised as follows:

1. Find the PCA decomposition of the data : $Y = A_P S_P$ where $A_P = U D ; S_P = V^T$ with $Y = U D V^T$ being the SVD.
2. Reduce the dimensionality by selecting the largest $M$ components of the PCA (thresholding the power - given by $D^2$), giving $Y_R = S_1$.
3. Find the orthogonal matrix $Q$ such that $f(S_2)$ is minimised where $Y_R = Q S_2$. The function $f(\cdot)$ needs to measure statistical dependence of the rows of $S_2$ (e.g. Negentropy).
4. The resulting rows of the matrix $S_2$ are the ICs (spatial maps) which are orthonormal and the columns of the matrix $A_2 = A_1 Q$ are the associated time courses in the original data space (which are not orthogonal in general, although in the reduced data space the columns of $Q$ are orthogonal).