Normal principal component analysis finds the direction in which the variation --- the spread of data points --- is largest, then it finds the orthogonal direction with the second largest variation. This continue until the dimension in the space is used up. The direction describes a new coordinate system which the data points can be referred to. The PCA is thus a rotation to new coordinates and a sorting with respect to the variance: A shift of basis.[54]
In the case of brain image analysis the dimension of the total space is much larger than the signal space. One scan describes a point the total space. With for example 100 scans the span of the empirical signal space is ''only'' 100 dimensional. SVD-PCA performs the usual PCA within that little space.
There is no second order dependencies among the principal components, but there can be higher order dependencies. These will be passed on to the neural network, and the neural network can, as a nonlinear modeler, use these dependencies.
Figure 4.3: Nonlinear data with the principal component: The two principal
component is dependent.
There is no data reduction involved in the SVD-PCA, but with the sorting that is included in the PCA, the higher principal components will contain little information, so that the data could be reduced.
The SVD-PCA can be said to produce a distributed representation of the input, compared to a localized: Each input pattern is represented in every component [27].
The SVD-PCA transformation is linear, so to handle data that ''is affine'' the volume must first be centered:
I will assume that the centering has been done, and call
the centered volume 
form now on.
