Geodesic distance on optimally regularized functional connectomes uncovers individual fingerprints.

Background: 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. Methods: 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 FCs, 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 functional magnetic resonance imaging scanning length and the parcellation used, it is not always attainable and sometimes a regularization procedure is required. Results: 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 data set-dependent and varies as a function of condition, parcellation, scanning length, and the number of frames used to compute the FCs. Discussion: 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 data set 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. Impact statement Functional connectomes (FCs) have a reproducible individual fingerprint, making it possible to study neurological and psychiatric phenomena at an individual level. But this requires an accurate way to compare FCs to establish individual-level associations with behavior and/or cognition. Although the canonical methods of comparing FCs (e.g., correlation, Euclidean) are adequate, geodesic distance provides a more principled and accurate way of comparing FCs by utilizing the underlying non-Euclidean geometry of correlation matrices. We demonstrate that by combining geodesic distance with an optimal amount of regularization, we can get substantially more reliable estimates of relative distances between FCs and thus uncover individual-level differences.