# Functional Connectome Fingerprinting Through Tucker Tensor Decomposition
###### Abstract
Using the fact that FCs estimated as correlation matrices lie on or inside a Symmetric Positive Definite (SPD) manifold, [PERSON] and colleagues proposed using geodesic distance to compare FCs [14]. In a follow-up study [12], the authors explored how optimally regularized FCs maximize the individual fingerprint of participants, measured by the geodesic distance between their FCs. A limitation of this approach is that the geodesic distance between FCs of size \(M\times M\) is a map \(\mathbb{R}^{M\times M}\times\mathbb{R}^{M\times M}\to\mathbb{R}\). Hence, even though the geodesic distance provides a global measure of similarity between FCs, it does not directly highlight the specific features that make the individual's functional connectivity unique. As an alternative to using FCs, tangent FCs have demonstrated a high capacity to predict cognition and behavior [15; 16; 17]. Most recently, ref. [13] analyzed the effects of tangent FCs with respect to fingerprinting. In that study, a high degree of fingerprinting was achieved not only across sessions of a unique participant, but also for matching the sessions of twins. To simultaneously overcome the drawbacks of the studies mentioned above, we propose utilizing tensor decomposition for FC fingerprinting. Tensor decomposition enables projecting high-dimensional data into a lower-dimensional space, preserving its structure while independently extracting meaningful information from each dimension. Tensors are multidimensional arrays with applications in various fields, including signal processing, computer vision, and neuroscience [18]. In brain connectomics, tensors enable modeling and analyzing the functional and structural connections within the brain by reducing the dimensionality of complex, interrelated, and high-dimensional data through tensor decomposition. In [19], the authors studied the dynamics of FCs to understand the process of formation and dissolution of brain functional networks through tensor decomposition techniques. In another study [20], it was demonstrated how the analysis of the tensor components enables the extraction of unbiased and interpretable descriptions of single-trial dynamics across many trials through low-dimensional representations of neural data. In [21], the authors discuss challenges associated with interpreting the brain connectivity patterns derived from tensor decomposition. To our knowledge, only [22] has considered using tensors in brain connectivity fingerprinting analysis in the literature. However, their study was based on structural connectivity. The task of identifying subjects through their functional connectomes has the additional challenge of dynamic changes and functional reconfigurations happening at a fast rate in response to cognitive stimuli. Through this study, the effectiveness of tensor based methods in uncovering FC fingerprints is assessed. The adopted fingerprinting framework can be broadly described as follows:  
1. Each participant underwent a total of sixteen data acquisition sessions, two for each of the considered fMRI conditions. Using the BOLD time series from each acquisition, an FC matrix of dimension Number of Brain Regions \(\times\) Number of Brain Regions is estimated, thus yielding a total of sixteen FC matrices for each participant. 2. For each condition, a tensor of dimensions Number of Brain Regions \(\times\) Number of Brain Regions \(\times\) Number of Participants, is constructed by concatenating all participants' FCs derived in the first acquisition session. Similarly, a second tensor is obtained for all conditions by concatenating all participants' FCs derived in the second acquisition session. 3. The obtained tensors are then decomposed via Tucker decomposition [23], yielding a core tensor and three factor matrices. The first two factor matrices contain cohort-level functional connectivity information and will later on be referred to as \"brain parcellation factor matrices\".