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## Section: New Results

### Population shrinkage of covariance (PoSCE) for better individual brain functional-connectivity estimation

Estimating covariances from functional Magnetic Resonance Imaging at rest (r-fMRI) can quantify interactions between brain regions. Also known as brain functional connectivity, it reflects inter-subject variations in behavior and cognition, and characterizes neuropathologies. Yet, with noisy and short time-series, as in r-fMRI, covariance estimation is challenging and calls for penalization, as with shrinkage approaches. We introduce population shrinkage of covariance estimator (PoSCE) : a covariance estimator that integrates prior knowledge of covariance distribution over a large population, leading to a non-isotropic shrinkage. The shrinkage is tailored to the Riemannian geometry of symmetric positive definite matrices. It is coupled with a probabilistic modeling of the individual and population covariance distributions. Experiments on two large r-fMRI datasets (HCP n=815, Cam-CAN n=626) show that PoSCE has a better bias-variance trade-off than existing covariance estimates: this estimator relates better functional-connectivity measures to cognition while capturing well intra-subject functional connectivity.

Figure 5. Tangent embedding and population prior modeling. ${\Sigma }_{0}$ is the mean covariance from a train set of covariances. It is the reference point in the tangent space. The population prior is defined as a Gaussian multivariate dis- tribution centered on $d{\Sigma }_{0}$ . ${\Lambda }_{0}$ is the covariance dispersion over the population. The arrows depict the mapping between the non-Euclidean covariance space and the tangent space.