Diffusion time-related structure-function coupling reveals differential association with inter-individual variations in body mass index

Body mass index (BMI) is an indicator of obesity, and recent neuroimaging studies have demonstrated that inter-individual variations in BMI are associated with altered brain structure and function. However, the mechanism underlying the alteration of structure-function correspondence according to BMI is under-investigated. In this study, we studied structural and functional connectivity derived from diffusion MRI tractography and inter-regional correlations of functional MRI time series, respectively. We combined the structural and functional connectivity information using the Riemannian optimization approach. First, the low-dimensional principal eigenvectors (i.e., gradients) of the structural connectivity were generated by applying diffusion map embedding with varying diffusion times. A transformation was identified so that the structural and functional embeddings share the same coordinate system, and subsequently, the functional connectivity matrix was simulated. Then, we generated gradients from the simulated functional connectivity matrix. We found the most apparent cortical hierarchical organization differentiating between low-level sensory and higher-order transmodal regions in the middle of the diffusion time, indicating that the hierarchical organization of the brain may reflect the intermediate mechanisms of mono- and polysynaptic communications. Associations between the functional gradients and BMI were strongest when the hierarchical structure was the most evident. Moreover, the gradient-BMI association map was related to the microstructural features, and the findings indicated that the BMI-related structure-function coupling was significantly associated with brain microstructure, particularly in higher-order transmodal areas. Finally, transcriptomic association analysis revealed the potential biological underpinnings specifying gene enrichment in the striatum, hypothalamus, and cortical cells. Our findings provide evidence that structure-function correspondence is strongly coupled with BMI when hierarchical organization is the most apparent and that the associations are related to the multiscale properties of the brain, leading to an advanced understanding of the neural mechanisms related to BMI.

U-SPDNet: An SPD manifold learning-based neural network for visual classification

With the development of neural networking techniques, several architectures for symmetric positive definite (SPD) matrix learning have recently been put forward in the computer vision and pattern recognition (CV&PR) community for mining fine-grained geometric features. However, the degradation of structural information during multi-stage feature transformation limits their capacity. To cope with this issue, this paper develops a U-shaped neural network on the SPD manifolds (U-SPDNet) for visual classification. The designed U-SPDNet contains two subsystems, one of which is a shrinking path (encoder) making up of a prevailing SPD manifold neural network (SPDNet (Huang and Van Gool, 2017)) for capturing compact representations from the input data. Another is a constructed symmetric expanding path (decoder) to upsample the encoded features, trained by a reconstruction error term. With this design, the degradation problem will be gradually alleviated during training. To enhance the representational capacity of U-SPDNet, we also append skip connections from encoder to decoder, realized by manifold-valued geometric operations, namely Riemannian barycenter and Riemannian optimization. On the MDSD, Virus, FPHA, and UAV-Human datasets, the accuracy achieved by our method is respectively 6.92%, 8.67%, 1.57%, and 1.08% higher than SPDNet, certifying its effectiveness.

Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators

A variational formulation for accelerated optimization on normed vector spaces was recently introduced in Wibisono et al. (PNAS 113:E7351-E7358, 2016), and later generalized to the Riemannian manifold setting in Duruisseaux and Leok (SJMDS, 2022a). This variational framework was exploited on normed vector spaces in Duruisseaux et al. (SJSC 43:A2949-A2980, 2021) using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization, and it was observed that geometric discretizations which respect the time-rescaling invariance and symplecticity of the Lagrangian and Hamiltonian flows were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. As such, it is natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In this paper, we consider the case of Riemannian manifolds embedded in a Euclidean space that can be characterized as the level set of a submersion. We will explore how holonomic constraints can be incorporated in discrete variational integrators to constrain the numerical discretization of the Riemannian Hamiltonian system to the Riemannian manifold, and we will test the performance of the resulting algorithms by solving eigenvalue and Procrustes problems formulated as optimization problems on the unit sphere and Stiefel manifold.

Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.

A Riemannian Optimization Algorithm for Joint Maximum Likelihood Estimation of High-Dimensional Exploratory Item Factor Analysis

There has been regained interest in joint maximum likelihood (JML) estimation of item factor analysis (IFA) recently, primarily due to its efficiency in handling high-dimensional data and numerous latent factors. It has been established under mild assumptions that the JML estimator is consistent as both the numbers of respondents and items tend to infinity. The current work presents an efficient Riemannian optimization algorithm for JML estimation of exploratory IFA with dichotomous response data, which takes advantage of the differential geometry of the fixed-rank matrix manifold. The proposed algorithm takes substantially less time to converge than a benchmark method that alternates between gradient ascent steps for person and item parameters. The performance of the proposed algorithm in the recovery of latent dimensionality, response probabilities, item parameters, and factor scores is evaluated via simulations.