Using distance on the Riemannian manifold to compare representations in brain and in models

Representational similarity analysis (RSA) summarizes activity patterns for a set of experimental conditions into a matrix composed of pairwise comparisons between activity patterns. Two examples of such matrices are the condition-by-condition inner product and correlation matrix. These representational matrices reside on the manifold of positive semidefinite matrices, called the Riemannian manifold. We hypothesize that representational similarities would be more accurately quantified by considering the underlying manifold of the representational matrices. Thus, we introduce the distance on the Riemannian manifold as a metric for comparing representations. Analyzing simulated and real fMRI data and considering a wide range of metrics, we show that the Riemannian distance is least susceptible to sampling bias, results in larger intra-subject reliability, and affords searchlight mapping with high sensitivity and specificity. Furthermore, we show that the Riemannian distance can be used for measuring multi-dimensional connectivity. This measure captures both univariate and multivariate connectivity and is also more sensitive to nonlinear regional interactions compared to the state-of-the-art measures. Applying our proposed metric to neural network representations of natural images, we demonstrate that it also possesses outstanding performance in quantifying similarity in models. Taken together, our results lend credence to the proposition that RSA should consider the manifold of the representational matrices to summarize response patterns in the brain and in models.