Infinite von Mises-Fisher Mixture Modeling of Whole Brain fMRI Data

Cluster analysis of functional magnetic resonance imaging (fMRI) data is often performed using gaussian mixture models, but when the time series are standardized such that the data reside on a hypersphere, this modeling assumption is questionable. The consequences of ignoring the underlying spherical manifold are rarely analyzed, in part due to the computational challenges imposed by directional statistics. In this letter, we discuss a Bayesian von Mises–Fisher (vMF) mixture model for data on the unit hypersphere and present an efficient inference procedure based on collapsed Markov chain Monte Carlo sampling. Comparing the vMF and gaussian mixture models on synthetic data, we demonstrate that the vMF model has a slight advantage inferring the true underlying clustering when compared to gaussian-based models on data generated from both a mixture of vMFs and a mixture of gaussians subsequently normalized. Thus, when performing model selection, the two models are not in agreement. Analyzing multisubject whole brain resting-state fMRI data from healthy adult subjects, we find that the vMF mixture model is considerably more reliable than the gaussian mixture model when comparing solutions across models trained on different groups of subjects, and again we find that the two models disagree on the optimal number of components. The analysis indicates that the fMRI data support more than a thousand clusters, and we confirm this is not a result of overfitting by demonstrating better prediction on data from held-out subjects. Our results highlight the utility of using directional statistics to model standardized fMRI data and demonstrate that whole brain segmentation of fMRI data requires a very large number of functional units in order to adequately account for the discernible statistical patterns in the data. </jats:p

Deep Adaptive Graph Clustering via Von Mises-Fisher Distributions

Graph clustering has been a hot research topic and is widely used in many fields, such as community detection in social networks. Lots of works combining auto-encoder and graph neural networks have been applied to clustering tasks by utilizing node attributes and graph structure. These works usually assumed the inherent parameters (i.e., size and variance) of different clusters in the latent embedding space are homogeneous, and hence the assigned probability is monotonous over the Euclidean distance between node embeddings and centroids. Unfortunately, this assumption usually does not hold since the size and concentration of different clusters can be quite different, which limits the clustering accuracy. In addition, the node embeddings in deep graph clustering methods are usually L2 normalized so that it lies on the surface of a unit hyper-sphere. To solve this problem, we proposed Deep Adaptive Graph Clustering via von Mises-Fisher distributions, namely DAGC. DAGC assumes the node embeddings H can be drawn from a von Mises-Fisher distribution and each cluster k is associated with cluster inherent parameters ρk which includes cluster center μ and cluster cohesion degree κ. Then we adopt an EM-like approach (i.e., (H|ρ) and (ρ|H), respectively) to learn the embedding and cluster inherent parameters alternately. Specifically, with the node embeddings, we proposed to update the cluster centers in an attraction-repulsion manner to make the cluster centers more separable. And given the cluster inherent parameters, a likelihood-based loss is proposed to make node embeddings more concentrated around cluster centers. Thus, DAGC can simultaneously improve the intra-cluster compactness and inter-cluster heterogeneity. Finally, extensive experiments conducted on four benchmark datasets have demonstrated that the proposed DAGC consistently outperforms the state-of-the-art methods, especially on imbalanced datasets

Geodesic finite mixture models

We present a novel approach for learning a finite mixture model on a Riemannian manifold in which Euclidean metrics are not applicable and one needs to resort to geodesic distances consistent with the manifold geometry. For this purpose, we draw inspiration on a variant of the expectation-maximization algorithm, that uses a minimum message length criterion to automatically estimate the optimal number of components from multivariate data lying on an Euclidean space. In order to use this approach on Riemannian manifolds, we propose a formulation in which each component is defined on a different tangent space, thus avoiding the problems associated with the loss of accuracy produced when linearizing the manifold with a single tangent space. Our approach can be applied to any type of manifold for which it is possible to estimate its tangent space. In particular, we show results on synthetic examples of a sphere and a quadric surface and on a large and complex dataset of human poses, where the proposed model is used as a regression tool for hypothesizing the geometry of occluded parts of the body.Peer ReviewedPostprint (published version