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Multivariate analysis of fMRI data has benefited substantially from advances in machine learning. Most recently, a range of probabilistic latent variable models applied to fMRI data have been successful in a variety of tasks, including identifying similarity patterns in neural data (Representational Similarity Analysis and its empirical Bayes variant, RSA and BRSA; Intersubject Functional Connectivity, ISFC), combining multi-subject datasets (Shared Response Mapping; SRM), and mapping between brain and behavior (Joint Modeling). Although these methods share some underpinnings, they have been developed as distinct methods, with distinct algorithms and software tools. We show how the matrix-variate normal (MN) formalism can unify some of these methods into a single framework. In doing so, we gain the ability to reuse noise modeling assumptions, algorithms, and code across models. Our primary theoretical contribution shows how some of these methods can be written as instantiations of the same model, allowing us to generalize them to flexibly modeling structured noise covariances. Our formalism permits novel model variants and improved estimation strategies: in contrast to SRM, the number of parameters for MN-SRM does not scale with the number of voxels or subjects; in contrast to BRSA, the number of parameters for MN-RSA scales additively rather than multiplicatively in the number of voxels. We empirically demonstrate advantages of two new methods derived in the formalism: for MN-RSA, we show up to 10x improvement in runtime, up to 6x improvement in RMSE, and more conservative behavior under the null. For MN-SRM, our method grants a modest improvement to out-of-sample reconstruction while relaxing an orthonormality constraint of SRM. We also provide a software prototyping tool for MN models that can flexibly reuse noise covariance assumptions and algorithms across models.
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Arterial Spin Labelling (ASL) functional Magnetic Resonance Imaging (fMRI) data provides a quantitative measure of blood perfusion, that can be correlated to neuronal activation. In contrast to BOLD measure, it is a direct measure of cerebral blood flow. However, ASL data has a lower SNR and resolution so that the recovery of the perfusion response of interest suffers from the contamination by a stronger hemodynamic component in the ASL signal. In this work we consider a model of both hemodynamic and perfusion components within the ASL signal. A physiological link between these two components is analyzed and used for a more accurate estimation of the perfusion response function in particular in the usual ASL low SNR conditions.
This note is concerned with accurate and computationally efficient approximations of moments of Gaussian random variables passed through sigmoid or softmax mappings. These approximations are semi-analytical (i.e. they involve the numerical adjustment of parametric forms) and highly accurate (they yield 5% error at most). We also highlight a few niche applications of these approximations, which arise in the context of, e.g., drift-diffusion models of decision making or non-parametric data clustering approaches. We provide these as examples of efficient alternatives to more tedious derivations that would be needed if one was to approach the underlying mathematical issues in a more formal way. We hope that this technical note will be helpful to modellers facing similar mathematical issues, although maybe stemming from different academic prospects.
Multi-modal brain functional connectivity (FC) data have shown great potential for providing insights into individual variations in behavioral and cognitive traits. The joint learning of multi-modal imaging data can utilize the intrinsic association, and thus can boost the learning performance. Although several multi-task based learning models have already been proposed by viewing the feature learning on each modality as one task, most of them ignore the geometric structure information inherent in the modalities, which may play an important role in extracting discriminative features. In this paper, we propose a new manifold regularized multi-task learning model by simultaneously considering between-subject and between-modality relationships. Besides employing a group-sparsity regularizer to jointly select a few common features across multiple tasks (modalities), we design a novel manifold regularizer to preserve the structure information both within and between modalities in our model. This will make our model more adaptive for realistic data analysis. Our model is then validated on the Philadelphia Neurodevelopmental Cohort dataset, where we regard our modalities as functional MRI (fMRI) data collected under two paradigms. Specifically, we conduct experimental studies on fMRI based FC network data in two task conditions for intelligence quotient (IQ) prediction. The results demonstrate that our proposed model can not only achieve improved prediction performance, but also yield a set of IQ-relevant biomarkers.
So-called sparse estimators arise in the context of model fitting, when one a priori assumes that only a few (unknown) model parameters deviate from zero. Sparsity constraints can be useful when the estimation problem is under-determined, i.e. when number of model parameters is much higher than the number of data points. Typically, such constraints are enforced by minimizing the L1 norm, which yields the so-called LASSO estimator. In this work, we propose a simple parameter transform that emulates sparse priors without sacrificing the simplicity and robustness of L2-norm regularization schemes. We show how L1 regularization can be obtained with a sparsify remapping of parameters under normal Bayesian priors, and we demonstrate the ensuing variational Laplace approach using Monte-Carlo simulations.
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. In addition, varieties can be used to model a richer class of nonlinear quadratic and higher degree curves and surfaces. We study the sampling requirements for matrix completion under a variety model with a focus on a union of affine subspaces. We also propose an efficient matrix completion algorithm that minimizes a convex or non-convex surrogate of the rank of the matrix of monomial features. Our algorithm uses the well-known kernel trick to avoid working directly with the high-dimensional monomial matrix. We show the proposed algorithm is able to recover synthetically generated data up to the predicted sampling complexity bounds. The proposed algorithm also outperforms standard low rank matrix completion and subspace clustering techniques in experiments with real data.
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