Quantitative measures

The low-dimensional embeddings, and therefore the color-coded dictionary maps, were validated using measures based on standard quantitative measures described in literature [8]. This was done in three steps. First, matrix S, describing the similarity of each dictionary entry with respect to all the other entries, was calculated for all the dictionaries. Since the Euclidean distance was used to optimize the t-SNE embeddings, this similarity matrix was defined as

Note that for SVD the embedding entries di were first normalized. The similarity matrices were normalized to a maximum value of unity on its diagonal by rescaling S according to Sij=1-Sij/maxij(Sij). Second, only the entries in the similarity matrix corresponding to WM (T2=80ms) and GM (T2=110ms) were selected [18], resulting in two similarity matrices of reduced size: SWM ∈R64×64 and SGM ∈R63×63. Note that these similarity matrices describe the encoding along one vertical line (of constant T2 values) in the color-coded dictionary maps, similar to those presented in Ref. [19]. Third, the distance between the similarity matrix and the identity matrix was used as a quantitative measure for WM and for GM:

with M=64 and N=63 being the number of elements in each dimension of the corresponding similarity matrices. We use the normalized ℓ1 norm to assign a single number to the encoding capability of an embedding for a specific T2 value. Note that these measures are low in case of a strong diagonal structure of Si, indicating a good encoding capability for tissue i.