Functional connectomes , which are symmetric correlation matrices, have been used to study changes in URLanization with aging, cognitive abilities and across a wide range of brain disorders. They have also been shown to provide a reproducible individual fingerprint, which has opened the possibility of personalized medicine for neuro/psychiatric disorders. Thus, an accurate way to compare functional connectomes is essential . Canonically, functional connectomes are compared using Pearson's correlation coefficient of the entire functional connectivity profiles. Recently, Venkatesh and colleagues proposed the use of geodesic distance as a more accurate way of comparing functional connectomes, one which reflects the underlying non-Euclidean geometry of the data. They evaluated their approach in the context of participant identification (or fingerprinting) and showed that it leads to higher identification rates for all functional MRI conditions assessed. Computing geodesic distance requires that the functional connectivity matrices being compared are invertible (or positive-definite ) . As this requirement depends on the fMRI scanning length and the parcellation used, it is not always attainable . Hence Venkatesh and colleagues employed a simple regularization procedure to ensure that the matrices were invertible . In the present technical note , we show that regularization is not only crucial for making the functional connectomes invertible, but also that an optimal magnitude of regularization can lead to systematically higher identification rates . We also show that this optimal value of regularization is highly dataset-dependent, and varies as a function of condition, parcellation,  and the number of frames used to compute the functional connectomes .
Functional connectomes (FCs), have been shown to provide a reproducible individual fingerprint, which has opened the possibility of personalized medicine for neuro/psychiatric disorders. Thus, developing accurate ways to compare FCs is essential to establish associations with behavior and/or cognition at the individual-level . Canonically, FCs are compared using Pearson\'s correlation coefficient of the entire functional connectivity profiles. Recently, it has been proposed that the use of geodesic distance is a more accurate way of comparing functional connectomes, one which reflects the underlying non-Euclidean geometry of the data. Computing geodesic distance requires FCs to be positive-definite and hence invertible matrices . As this requirement depends on the fMRI scanning length and the parcellation used, it is not always attainable and sometimes a regularization procedure is required . In the present work , we show that regularization is not only an algebraic operation for making FCs invertible, but also that an optimal magnitude of regularization leads to systematically higher fingerprints . We also show evidence that optimal regularization is dataset-dependent, and varies as a function of condition, parcellation, scanning length, and the number of frames used to compute the FCs. We demonstrate that a universally fixed regularization does not fully uncover the potential of geodesic distance on individual fingerprinting, and indeed could severely diminish it. Thus, an optimal regularization must be estimated on each dataset to uncover the most differentiable across-subject and reproducible within-subject geodesic distances between FCs. The resulting pairwise geodesic distances at the optimal regularization level constitute a very reliable quantification of differences between subjects .
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