Section: New Results

Population-shrinkage of covariance to estimate better brain functional connectivity

Brain functional connectivity, obtained from functional Magnetic Resonance Imaging at rest (r-fMRI), reflects inter-subject variations in behavior and characterizes neuropathologies. It is captured by the covariance matrix between time series of remote brain regions. With noisy and short time series as in r-fMRI, covariance estimation calls for penalization, and shrinkage approaches are popular. Here we introduce a new covariance estimator based on a non-isotropic shrinkage that integrates prior knowledge of the covariance distribution over a large population. The estimator performs shrinkage tailored to the Riemannian geometry of symmetric positive definite matrices, coupled with a probabilistic modeling of the subject and population covariance distributions. Experiments on a large-scale dataset show that such estimators resolve better intra-and inter-subject functional connectivities compared existing co-variance estimates. We also demonstrate that the estimator improves the relationship across subjects between their functional-connectivity measures and their behavioral assessments. More information can be found in Fig. 4 in [47].

Figure 4. (a) Shrunk embedding estimation workflow: the empirical covariance is estimated from r-fMRI time-series; it is projected onto a tangent space built from a prior population; the embedding is then shrunk towards the prior $({\overrightarrow{𝐝\Sigma }}_0,{\overrightarrow{\overrightarrow{\Lambda }}}_0)$. (b) Principle of tangent embedding shrinkage towards population distribution.