Transport on Riemannian Manifold for Connectivity-Based Brain Decoding

Abstract

Abstract-There is a recent interest in using functional magnetic resonance imaging (fMRI) for decoding more naturalistic, cognitive states, in which subjects perform various tasks in a continuous, self-directed manner. In this setting, the set of brain volumes over the entire task duration is usually taken as a single sample with connectivity estimates, such as Pearson's correlation, employed as features. Since covariance matrices live on the positive semidefinite cone, their elements are inherently inter-related. The assumption of uncorrelated features implicit in most classifier learning algorithms is thus violated. Coupled with the usual small sample sizes, the generalizability of the learned classifiers is limited, and the identification of significant brain connections from the classifier weights is nontrivial. In this paper, we present a Riemannian approach for connectivity-based brain decoding. The core idea is to project the covariance estimates onto a common tangent space to reduce the statistical dependencies between their elements. For this, we propose a matrix whitening transport, and compare it against parallel transport implemented via the Schild's ladder algorithm. To validate our classification approach, we apply it to fMRI data acquired from twenty four subjects during four continuous, selfdriven tasks. We show that our approach provides significantly higher classification accuracy than directly using Pearson's correlation and its regularized variants as features. To facilitate result interpretation, we further propose a non-parametric scheme that combines bootstrapping and permutation testing for identifying significantly discriminative brain connections from the classifier weights. Using this scheme, a number of neuroanatomically meaningful connections are detected, whereas no significant connections are found with pure permutation testing.

Optimising network modelling methods for fMRI

“…To do so, Riemannian space is approximated with an associated tangent space and then functional connectivity estimates are mapped to the tangent space. We have reported in this paper that this tangent space parameterization results in a significant increase in predictive power of functional connectivity estimates, as previously shown in [83,84]. Our results have also demonstrated that additional shrinkage is not necessarily required on well regularized connectivity estimates in tangent space; however, could be applied for noiser data.…”

“…In terms of step-1 (brain parcellation), we compared sICA with MODL, and found that the performance of these is similar (although sICA outperforms for age prediction, as shown in Figure A.34). Lastly, we compared ICA 50D with YEO higher dimensional parcellation (200D and 400D) on HCP data, and found that low dimensional sICA still outperforms higher dimensional YEO parcellation (as shown in Comparison with Literature: Somewhat similar evaluations of various methods (in this problem domain) have previously been carried out in [83,84]. In [83], authors proposed that matrix whitening transform and parallel transport could be utilized to project covariance matrices into a common tangent space and evaluated this method on twenty four healthy subjects.…”

“…Lastly, we compared ICA 50D with YEO higher dimensional parcellation (200D and 400D) on HCP data, and found that low dimensional sICA still outperforms higher dimensional YEO parcellation (as shown in Comparison with Literature: Somewhat similar evaluations of various methods (in this problem domain) have previously been carried out in [83,84]. In [83], authors proposed that matrix whitening transform and parallel transport could be utilized to project covariance matrices into a common tangent space and evaluated this method on twenty four healthy subjects. These former results build a foundation for estimation of functional connectivity using manifold operations, as higher prediction accuracy was achieved using tangent space parameterization.…”

“…In this paper, we evaluate an arguably more principled approach to represent correlation-based functional connectivity using Riemannian geometry [27,28]. This method is founded upon mathematical principles which involve manipulating correlation-based estimates in the tangential space of Riemannian manifold [30] where Euclidean geometry can be applied.…”

Multi-Link Analysis: Brain Network Comparison via Sparse Connectivity Analysis

“…While, Chen et al 18 enhanced NBS regulating the topological structures comprised. Other research groups [19][20][21] leveraged support vector machines (SVM) weights to identify discriminating regions. SVM is a supervised learning method which constructs a hyperplane or set of hyperplanes in a high-or infinite-dimensional space used for classification.…”