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Zhengwu Zhang

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Applications Computer Vision Machine Learning Methodology Computation math.FA Neurons and Cognition

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preprint2026arXiv

Diffeomorphic Cortical Alignment via Direct Warping of Streamline Endpoints

Cortical surface registration is often driven by local geometric descriptors (e.g., sulcal depth and curvature). While this approach achieves geometric correspondence, it neglects the long-range wiring constraints imposed by white-matter anatomy. Diffusion MRI tractography offers these crucial constraints; however, prior connectivity-informed pipelines typically align precomputed connectivity matrices, making the optimization highly sensitive to connectivity estimation and its resolution. In this paper, we introduce a novel connectivity-based surface registration method that aligns cortical surfaces by operating directly on white-matter fiber-tract endpoints. We model tract endpoints as a point cloud on the product manifold $Ω\times Ω$, where $Ω$ represents the spherical domain of the inflated cortical hemispheres. Our alignment method iteratively (i) computes a small diffeomorphic warp for $Ω$ by minimizing connectivity mismatch, and (ii) updates the endpoints based on this warp. The method relies on a geometric framework that ensures output warps are diffeomorphisms and has a final goal that optimizes the matching of well-known fiber bundles. Experiments on Human Connectome Project (HCP) data demonstrate improved tract-level correspondence, achieving higher connectivity-level overlap coefficients on major fiber bundles and stronger robustness across grid resolutions for $Ω$ compared to state-of-the-art methods such as ENCORE and MSMAll.

preprint2025arXiv

NeuroPMD: Neural Fields for Density Estimation on Product Manifolds

We propose a novel deep neural network methodology for density estimation on product Riemannian manifold domains. In our approach, the network directly parameterizes the unknown density function and is trained using a penalized maximum likelihood framework, with a penalty term formed using manifold differential operators. The network architecture and estimation algorithm are carefully designed to handle the challenges of high-dimensional product manifold domains, effectively mitigating the curse of dimensionality that limits traditional kernel and basis expansion estimators, as well as overcoming the convergence issues encountered by non-specialized neural network methods. Extensive simulations and a real-world application to brain structural connectivity data highlight the clear advantages of our method over the competing alternatives.

preprint2022arXiv

Amplitude Mean of Functional Data on $\mathbb{S}^2$

Manifold-valued functional data analysis (FDA) recently becomes an active area of research motivated by the raising availability of trajectories or longitudinal data observed on non-linear manifolds. The challenges of analyzing such data come from many aspects, including infinite dimensionality and nonlinearity, as well as time-domain or phase variability. In this paper, we study the amplitude part of manifold-valued functions on $\mathbb{S}^2$, which is invariant to random time warping or re-parameterization. Utilizing the nice geometry of $\mathbb{S}^2$, we develop a set of efficient and accurate tools for temporal alignment of functions, geodesic computing, and sample mean calculation. At the heart of these tools, they rely on gradient descent algorithms with carefully derived gradients. We show the advantages of these newly developed tools over its competitors with extensive simulations and real data and demonstrate the importance of considering the amplitude part of functions instead of mixing it with phase variability in manifold-valued FDA.

preprint2022arXiv

Analyzing Brain Structural Connectivity as Continuous Random Functions

This work considers a continuous framework to characterize the population-level variability of structural connectivity. Our framework assumes the observed white matter fiber tract endpoints are driven by a latent random function defined over a product manifold domain. To overcome the computational challenges of analyzing such complex latent functions, we develop an efficient algorithm to construct a data-driven reduced-rank function space to represent the latent continuous connectivity. Using real data from the Human Connectome Project, we show that our method outperforms state-of-the-art approaches applied to the traditional atlas-based structural connectivity matrices on connectivity analysis tasks of interest. We also demonstrate how our method can be used to identify localized regions and connectivity patterns on the cortical surface associated with significant group differences. Code will be made available at https://github.com/sbci-brain.

preprint2022arXiv

Outlier Detection for Multi-Network Data

It has become routine in neuroscience studies to measure brain networks for different individuals using neuroimaging. These networks are typically expressed as adjacency matrices, with each cell containing a summary of connectivity between a pair of brain regions. There is an emerging statistical literature describing methods for the analysis of such multi-network data in which nodes are common across networks but the edges vary. However, there has been essentially no consideration of the important problem of outlier detection. In particular, for certain subjects, the neuroimaging data are so poor quality that the network cannot be reliably reconstructed. For such subjects, the resulting adjacency matrix may be mostly zero or exhibit a bizarre pattern not consistent with a functioning brain. These outlying networks may serve as influential points, contaminating subsequent statistical analyses. We propose a simple Outlier DetectIon for Networks (ODIN) method relying on an influence measure under a hierarchical generalized linear model for the adjacency matrices. An efficient computational algorithm is described, and ODIN is illustrated through simulations and an application to data from the UK Biobank. ODIN was successful in identifying moderate to extreme outliers. Removing such outliers can significantly change inferences in downstream applications.

preprint2020arXiv

Bayesian Hierarchical Modeling on Covariance Valued Data

Analysis of structural and functional connectivity (FC) of human brains is of pivotal importance for diagnosis of cognitive ability. The Human Connectome Project (HCP) provides an excellent source of neural data across different regions of interest (ROIs) of the living human brain. Individual specific data were available from an existing analysis (Dai et al., 2017) in the form of time varying covariance matrices representing the brain activity as the subjects perform a specific task. As a preliminary objective of studying the heterogeneity of brain connectomics across the population, we develop a probabilistic model for a sample of covariance matrices using a scaled Wishart distribution. We stress here that our data units are available in the form of covariance matrices, and we use the Wishart distribution to create our likelihood function rather than its more common usage as a prior on covariance matrices. Based on empirical explorations suggesting the data matrices to have low effective rank, we further model the center of the Wishart distribution using an orthogonal factor model type decomposition. We encourage shrinkage towards a low rank structure through a novel shrinkage prior and discuss strategies to sample from the posterior distribution using a combination of Gibbs and slice sampling. We extend our modeling framework to a dynamic setting to detect change points. The efficacy of the approach is explored in various simulation settings and exemplified on several case studies including our motivating HCP data. We extend our modeling framework to a dynamic setting to detect change points.

preprint2019arXiv

Discovering Common Change-Point Patterns in Functional Connectivity Across Subjects

This paper studies change-points in human brain functional connectivity (FC) and seeks patterns that are common across multiple subjects under identical external stimulus. FC relates to the similarity of fMRI responses across different brain regions when the brain is simply resting or performing a task. While the dynamic nature of FC is well accepted, this paper develops a formal statistical test for finding {\it change-points} in times series associated with FC. It represents short-term connectivity by a symmetric positive-definite matrix, and uses a Riemannian metric on this space to develop a graphical method for detecting change-points in a time series of such matrices. It also provides a graphical representation of estimated FC for stationary subintervals in between the detected change-points. Furthermore, it uses a temporal alignment of the test statistic, viewed as a real-valued function over time, to remove inter-subject variability and to discover common change-point patterns across subjects. This method is illustrated using data from Human Connectome Project (HCP) database for multiple subjects and tasks.

preprint2016arXiv

Low-Rank Representation over the Manifold of Curves

In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e.\ each data point is a function of some variable such as time and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function. In this paper we propose a method to analyse subspace structure of the functional data by using the state of the art Low-Rank Representation (LRR). Experimental evaluation on synthetic and real data reveals that this method massively outperforms conventional LRR in tasks concerning functional data.

preprint2016arXiv

Nonparametric Bayes Models of Fiber Curves Connecting Brain Regions

In studying structural inter-connections in the human brain, it is common to first estimate fiber bundles connecting different regions of the brain relying on diffusion MRI. These fiber bundles act as highways for neural activity and communication, snaking through the brain and connecting different regions. Current statistical methods for analyzing these fibers reduce the rich information into an adjacency matrix, with the elements containing a count of the number of fibers or a mean diffusion feature (such as fractional anisotropy) along the fibers. The goal of this article is to avoid discarding the rich functional data on the shape, size and orientation of fibers, developing flexible models for characterizing the population distribution of fibers between brain regions of interest within and across different individuals. We start by decomposing each fiber in each individual's brain into a corresponding rotation matrix, shape and translation from a global reference curve. These components can be viewed as data lying on a product space composed of different Euclidean spaces and manifolds. To non-parametrically model the distribution within and across individuals, we rely on a hierarchical mixture of product kernels specific to the component spaces. Taking a Bayesian approach to inference, we develop an efficient method for posterior sampling. The approach automatically produces clusters of fibers within and across individuals, and yields interesting new insights into variation in fiber curves, while providing a useful starting point for more elaborate models relating fibers to covariates and neuropsychiatric traits.

preprint2016arXiv

Phase-Amplitude Separation and Modeling of Spherical Trajectories

This paper studies the problem of separating phase-amplitude components in sample paths of a spherical process (longitudinal data on a unit two-sphere). Such separation is essential for efficient modeling and statistical analysis of spherical longitudinal data in a manner that is invariant to any phase variability. The key idea is to represent each path or trajectory with a pair of variables, a starting point and a Transported Square-Root Velocity Curve (TSRVC). A TSRVC is a curve in the tangent (vector) space at the starting point and has some important invariance properties under the L2 norm. The space of all such curves forms a vector bundle and the L2 norm, along with the standard Riemannian metric on S2, provides a natural metric on this vector bundle. This invariant representation allows for separating phase and amplitude components in given data, using a template-based idea. Furthermore, the metric property is useful in deriving computational procedures for clustering, mean computation, principal component analysis (PCA), and modeling. This comprehensive framework is demonstrated using two datasets: a set of bird-migration trajectories and a set of hurricane paths in the Atlantic ocean.

preprint2015arXiv

Bayesian Clustering of Shapes of Curves

Unsupervised clustering of curves according to their shapes is an important problem with broad scientific applications. The existing model-based clustering techniques either rely on simple probability models (e.g., Gaussian) that are not generally valid for shape analysis or assume the number of clusters. We develop an efficient Bayesian method to cluster curve data using an elastic shape metric that is based on joint registration and comparison of shapes of curves. The elastic-inner product matrix obtained from the data is modeled using a Wishart distribution whose parameters are assigned carefully chosen prior distributions to allow for automatic inference on the number of clusters. Posterior is sampled through an efficient Markov chain Monte Carlo procedure based on the Chinese restaurant process to infer (1) the posterior distribution on the number of clusters, and (2) clustering configuration of shapes. This method is demonstrated on a variety of synthetic data and real data examples on protein structure analysis, cell shape analysis in microscopy images, and clustering of shaped from MPEG7 database.

preprint2015arXiv

Video-Based Action Recognition Using Rate-Invariant Analysis of Covariance Trajectories

Statistical classification of actions in videos is mostly performed by extracting relevant features, particularly covariance features, from image frames and studying time series associated with temporal evolutions of these features. A natural mathematical representation of activity videos is in form of parameterized trajectories on the covariance manifold, i.e. the set of symmetric, positive-definite matrices (SPDMs). The variable execution-rates of actions implies variable parameterizations of the resulting trajectories, and complicates their classification. Since action classes are invariant to execution rates, one requires rate-invariant metrics for comparing trajectories. A recent paper represented trajectories using their transported square-root vector fields (TSRVFs), defined by parallel translating scaled-velocity vectors of trajectories to a reference tangent space on the manifold. To avoid arbitrariness of selecting the reference and to reduce distortion introduced during this mapping, we develop a purely intrinsic approach where SPDM trajectories are represented by redefining their TSRVFs at the starting points of the trajectories, and analyzed as elements of a vector bundle on the manifold. Using a natural Riemannain metric on vector bundles of SPDMs, we compute geodesic paths and geodesic distances between trajectories in the quotient space of this vector bundle, with respect to the re-parameterization group. This makes the resulting comparison of trajectories invariant to their re-parameterization. We demonstrate this framework on two applications involving video classification: visual speech recognition or lip-reading and hand-gesture recognition. In both cases we achieve results either comparable to or better than the current literature.

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