Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets

Spatial process models for analyzing geostatistical data entail computations that become prohibitive as the number of spatial locations become large. This manuscript develops a class of highly scalable Nearest Neighbor Gaussian Process (NNGP) models to provide fully model-based inference for large geostatistical datasets. We establish that the NNGP is a well-defined spatial process providing legitimate finite-dimensional Gaussian densities with sparse precision matrices. We embed the NNGP as a sparsity-inducing prior within a rich hierarchical modeling framework and outline how computationally efficient Markov chain Monte Carlo (MCMC) algorithms can be executed without storing or decomposing large matrices. The floating point operations (flops) per iteration of this algorithm is linear in the number of spatial locations, thereby rendering substantial scalability. We illustrate the computational and inferential benefits of the NNGP over competing methods using simulation studies and also analyze forest biomass from a massive United States Forest Inventory dataset at a scale that precludes alternative dimension-reducing methods

Multi-objective point cloud autoencoders for explainable myocardial infarction prediction

Myocardial infarction (MI) is one of the most common causes of death in the world. Image-based biomarkers commonly used in the clinic, such as ejection fraction, fail to capture more complex patterns in the heart's 3D anatomy and thus limit diagnostic accuracy. In this work, we present the multi-objective point cloud autoencoder as a novel geometric deep learning approach for explainable infarction prediction, based on multi-class 3D point cloud representations of cardiac anatomy and function. Its architecture consists of multiple task-specific branches connected by a low-dimensional latent space to allow for effective multi-objective learning of both reconstruction and MI prediction, while capturing pathology-specific 3D shape information in an interpretable latent space. Furthermore, its hierarchical branch design with point cloud-based deep learning operations enables efficient multi-scale feature learning directly on high-resolution anatomy point clouds. In our experiments on a large UK Biobank dataset, the multi-objective point cloud autoencoder is able to accurately reconstruct multi-temporal 3D shapes with Chamfer distances between predicted and input anatomies below the underlying images' pixel resolution. Our method outperforms multiple machine learning and deep learning benchmarks for the task of incident MI prediction by 19% in terms of Area Under the Receiver Operating Characteristic curve. In addition, its task-specific compact latent space exhibits easily separable control and MI clusters with clinically plausible associations between subject encodings and corresponding 3D shapes, thus demonstrating the explainability of the prediction

Modeling 3D cardiac contraction and relaxation with point cloud deformation networks

Global single-valued biomarkers of cardiac function typically used in clinical practice, such as ejection fraction, provide limited insight on the true 3D cardiac deformation process and hence, limit the understanding of both healthy and pathological cardiac mechanics. In this work, we propose the Point Cloud Deformation Network (PCD-Net) as a novel geometric deep learning approach to model 3D cardiac contraction and relaxation between the extreme ends of the cardiac cycle. It employs the recent advances in point cloud-based deep learning into an encoder-decoder structure, in order to enable efficient multi-scale feature learning directly on multi-class 3D point cloud representations of the cardiac anatomy. We evaluate our approach on a large dataset of over 10,000 cases from the UK Biobank study and find average Chamfer distances between the predicted and ground truth anatomies below the pixel resolution of the underlying image acquisition. Furthermore, we observe similar clinical metrics between predicted and ground truth populations and show that the PCD-Net can successfully capture subpopulation-specific differences between normal subjects and myocardial infarction (MI) patients. We then demonstrate that the learned 3D deformation patterns outperform multiple clinical benchmarks by 13% and 7% in terms of area under the receiver operating characteristic curve for the tasks of prevalent MI detection and incident MI prediction and by 7% in terms of Harrell's concordance index for MI survival analysis

Graphical Gaussian Process Models for Highly Multivariate Spatial Data

For multivariate spatial Gaussian process (GP) models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence among the variables. This is undesirable, especially for highly multivariate settings, where popular cross-covariance functions such as the multivariate Mat\'ern suffer from a "curse of dimensionality" as the number of parameters and floating point operations scale up in quadratic and cubic order, respectively, in the number of variables. We propose a class of multivariate "Graphical Gaussian Processes" using a general construction called "stitching" that crafts cross-covariance functions from graphs and ensures process-level conditional independence among variables. For the Mat\'ern family of functions, stitching yields a multivariate GP whose univariate components are Mat\'ern GPs, and conforms to process-level conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Mat\'ern GP to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling

Graph-constrained Analysis for Multivariate Functional Data

Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model representing the conditional relationships between the functional variables. However, in many applications of multivariate functional data, the graph is known and existing functional GGM methods cannot preserve a given graphical constraint. In this manuscript, we demonstrate how to conduct multivariate functional analysis that exactly conforms to a given inter-variable graph. We first show the equivalence between partially separable functional GGM and graphical Gaussian processes (GP), proposed originally for constructing optimal covariance functions for multivariate spatial data that retain the conditional independence relations in a given graphical model. The theoretical connection help design a new algorithm that leverages Dempster's covariance selection to calculate the maximum likelihood estimate of the covariance function for multivariate functional data under graphical constraints. We also show that the finite term truncation of functional GGM basis expansion used in practice is equivalent to a low-rank graphical GP, which is known to oversmooth marginal distributions. To remedy this, we extend our algorithm to better preserve marginal distributions while still respecting the graph and retaining computational scalability. The insights obtained from the new results presented in this manuscript will help practitioners better understand the relationship between these graphical models and in deciding on the appropriate method for their specific multivariate data analysis task. The benefits of the proposed algorithms are illustrated using empirical experiments and an application to functional modeling of neuroimaging data using the connectivity graph among regions of the brain.Comment: 23 pages, 6 figure

Multi-class point cloud completion networks for 3D cardiac anatomy reconstruction from cine magnetic resonance images

Cine magnetic resonance imaging (MRI) is the current gold standard for the assessment of cardiac anatomy and function. However, it typically only acquires a set of two-dimensional (2D) slices of the underlying three-dimensional (3D) anatomy of the heart, thus limiting the understanding and analysis of both healthy and pathological cardiac morphology and physiology. In this paper, we propose a novel fully automatic surface reconstruction pipeline capable of reconstructing multi-class 3D cardiac anatomy meshes from raw cine MRI acquisitions. Its key component is a multi-class point cloud completion network (PCCN) capable of correcting both the sparsity and misalignment issues of the 3D reconstruction task in a unified model. We first evaluate the PCCN on a large synthetic dataset of biventricular anatomies and observe Chamfer distances between reconstructed and gold standard anatomies below or similar to the underlying image resolution for multiple levels of slice misalignment. Furthermore, we find a reduction in reconstruction error compared to a benchmark 3D U-Net by 32% and 24% in terms of Hausdorff distance and mean surface distance, respectively. We then apply the PCCN as part of our automated reconstruction pipeline to 1000 subjects from the UK Biobank study in a cross-domain transfer setting and demonstrate its ability to reconstruct accurate and topologically plausible biventricular heart meshes with clinical metrics comparable to the previous literature. Finally, we investigate the robustness of our proposed approach and observe its capacity to successfully handle multiple common outlier conditions