Overview of the Singular Values Decomposition (SVD) analysis in the context of FDG and Tau PET datasets. The technique performs an indirect SVD decomposition of the full voxel-by-voxel cross-correlation matrix by producing mutually orthogonal spatial components that summarizes underlying pattens of correlations.
Since the size of the whole brain cross-correlation matrix (let's call it "C") is usually much larger than its rank (i.e. many more voxels than subjects), the statistical inference over this matrix, or even its storage, becomes impractical.
As such, dimensionality reduction techniques based on matrix decompositions, like SVD, are required. In practice, C is approximated by the first few components, ordered according to their corresponding singular values. Some algebraic manipulations show that there is no need to construct or store the matrix C for extracting significant cross-correlations patterns given by the corresponding spatial loadings or eigenimages in each PET modality.
Within modality, voxels with high spatial loading values co-vary together (i.e. they are positively correlated), while voxels with high opposite signed values are negatively correlated. Thus, high spatial loadings of an eigenimage in one modality can be interpreted as a spatial network of highly correlated voxels that are, in turn, maximally cross-correlated (in the sense of canonical correlations) with the spatial network of voxels showing high values in the other modality.