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Recent results in coupled or temporal graphical models offer schemes for estimating the relationship structure between features when the data come from related (but distinct) longitudinal sources. A novel application of these ideas is for analyzing group-level differences, i.e., in identifying if trends of estimated objects (e.g., covariance or precision matrices) are different across disparate conditions (e.g., gender or disease). Often, poor effect sizes make detecting the differential signal over the full set of features difficult: for example, dependencies between only a subset of features may manifest differently across groups. In this work, we first give a parametric model for estimating trends in the space of SPD matrices as a function of one or more covariates. We then generalize scan statistics to graph structures, to search over distinct subsets of features (graph partitions) whose temporal dependency structure may show statistically significant groupwise differences. We theoretically analyze the Family Wise Error Rate (FWER) and bounds on Type 1 and Type 2 error. Evaluating on U.S. census data, we identify groups of states with cultural and legal overlap related to baby name trends and drug usage. On a cohort of individuals with risk factors for Alzheimer's disease (but otherwise cognitively healthy), we find scientifically interesting group differences where the default analysis, i.e., models estimated on the full graph, do not survive reasonable significance thresholds. [ABSTRACT FROM AUTHOR]

The application of modern machine learning to retinal image analyses offers valuable insights into a broad range of human health conditions beyond ophthalmic diseases. Additionally, data sharing is key to fully realizing the potential of machine learning models by providing a rich and diverse collection of training data. However, the personallyidentifying nature of retinal images, encompassing the unique vascular structure of each individual, often prevents this data from being shared openly. While prior works have explored image de-identification strategies based on synthetic averaging of images in other domains (e.g. facial images), existing techniques face difficulty in preserving both privacy and clinical utility in retinal images, as we demonstrate in our work. We therefore introduce k -SALSA, a generative adversarial network (GAN)-based framework for synthesizing retinal fundus images that summarize a given private dataset while satisfying the privacy notion of k -anonymity. k -SALSA brings together state-of-the-art techniques for training and inverting GANs to achieve practical performance on retinal images. Furthermore, k -SALSA leverages a new technique, called local style alignment, to generate a synthetic average that maximizes the retention of fine-grain visual patterns in the source images, thus improving the clinical utility of the generated images. On two benchmark datasets of diabetic retinopathy (EyePACS and APTOS), we demonstrate our improvement upon existing methods with respect to image fidelity, classification performance, and mitigation of membership inference attacks. Our work represents a step toward broader sharing of retinal images for scientific collaboration. Code is available at https://github.com/hcholab/k-salsa.

There has recently been a concerted effort to derive mechanisms in vision and machine learning systems to offer uncertainty estimates of the predictions they make. Clearly, there are benefits to a system that is not only accurate but also has a sense for when it is not. Existing proposals center around Bayesian interpretations of modern deep architectures - these are effective but can often be computationally demanding. We show how classical ideas in the literature on exponential families on probabilistic networks provide an excellent starting point to derive uncertainty estimates in Gated Recurrent Units (GRU). Our proposal directly quantifies uncertainty deterministically , without the need for costly sampling-based estimation. We show that while uncertainty is quite useful by itself in computer vision and machine learning, we also demonstrate that it can play a key role in enabling statistical analysis with deep networks in neuroimaging studies with normative modeling methods. To our knowledge, this is the first result describing sampling-free uncertainty estimation for powerful sequential models such as GRUs.

There is much interest in developing algorithms based on 3D convolutional neural networks (CNNs) for performing regression and classification with brain imaging data and more generally, with biomedical imaging data. A standard assumption in learning is that the training samples are independently drawn from the underlying distribution. In computer vision, where we have millions of training examples, this assumption is violated but the empirical performance may remain satisfactory. But in many biomedical studies with just a few hundred training examples, one often has multiple samples per participant and/or data may be curated by pooling datasets from a few different institutions. Here, the violation of the independent samples assumption turns out to be more significant, especially in small-to-medium sized datasets. Motivated by this need, we show how 3D CNNs can be modified to deal with dependent samples. We show that even with standard 3D CNNs, there is value in augmenting the network to exploit information regarding dependent samples. We present empirical results for predicting cognitive trajectories (slope and intercept) from morphometric change images derived from multiple time points. With terms which encode dependency between samples in the model, we get consistent improvements over a strong baseline which ignores such knowledge.

A sizable body of work on relative attributes provides evidence that relating pairs of images along a continuum of strength pertaining to a visual attribute yields improvements in a variety of vision tasks. In this paper, we show how emerging ideas in graph neural networks can yield a solution to various problems that broadly fall under relative attribute learning. Our main idea is the observation that relative attribute learning naturally benefits from exploiting the graph of dependencies among the different relative attributes of images, especially when only partial ordering is provided at training time. We use message passing to perform end to end learning of the image representations, their relationships as well as the interplay between different attributes. Our experiments show that this simple framework is effective in achieving competitive accuracy with specialized methods for both relative attribute learning and binary attribute prediction, while relaxing the requirements on the training data and/or the number of parameters, or both.

Performing large scale hypothesis testing on brain imaging data to identify group-wise differences (e.g., between healthy and diseased subjects) typically leads to a large number of tests (one per voxel). Multiple testing adjustment (or correction) is necessary to control false positives, which may lead to lower detection power in detecting true positives. Motivated by the use of so-called "independent filtering" techniques in statistics (for genomics applications), this paper investigates the use of independent filtering for manifold-valued data (e.g., Diffusion Tensor Imaging, Cauchy Deformation Tensors) which are broadly used in neuroimaging studies. Inspired by the concept of variance of a Riemannian Gaussian distribution, a type of non-specific data-dependent Riemannian variance filter is proposed. In practice, the filter will select a subset of the full set of voxels for performing the statistical test, leading to a more appropriate multiple testing correction. Our experiments on synthetic/simulated manifold-valued data show that the detection power is improved when the statistical tests are performed on the voxel locations that "pass" the filter. Given the broadening scope of applications where manifold-valued data are utilized, the scheme can serve as a general feature selection scheme.

Statistical machine learning models that operate on manifold-valued data are being extensively studied in vision, motivated by applications in activity recognition, feature tracking and medical imaging. While non-parametric methods have been relatively well studied in the literature, efficient formulations for parametric models (which may offer benefits in small sample size regimes) have only emerged recently. So far, manifold-valued regression models (such as geodesic regression) are restricted to the analysis of cross-sectional data, i.e., the so-called "fixed effects" in statistics. But in most "longitudinal analysis" (e.g., when a participant provides multiple measurements, over time) the application of fixed effects models is problematic. In an effort to answer this need, this paper generalizes non-linear mixed effects model to the regime where the response variable is manifold-valued, i.e., f : R d → ℳ. We derive the underlying model and estimation schemes and demonstrate the immediate benefits such a model can provide - both for group level and individual level analysis - on longitudinal brain imaging data. The direct consequence of our results is that longitudinal analysis of manifold-valued measurements (especially, the symmetric positive definite manifold) can be conducted in a computationally tractable manner.

Statistical models such as linear regression drive numerous applications in computer vision and machine learning. The landscape of practical deployments of these formulations is dominated by forward regression models that estimate the parameters of a function mapping a set of p covariates, x , to a response variable, y . The less known alternative, Inverse Regression, offers various benefits that are much less explored in vision problems. The goal of this paper is to show how Inverse Regression in the "abundant" feature setting (i.e., many subsets of features are associated with the target label or response, as is the case for images), together with a statistical construction called Sufficient Reduction, yields highly flexible models that are a natural fit for model estimation tasks in vision. Specifically, we obtain formulations that provide relevance of individual covariates used in prediction, at the level of specific examples/samples - in a sense, explaining why a particular prediction was made. With no compromise in performance relative to other methods, an ability to interpret why a learning algorithm is behaving in a specific way for each prediction, adds significant value in numerous applications. We illustrate these properties and the benefits of Abundant Inverse Regression (AIR) on three distinct applications.

A major goal of imaging studies such as the (ongoing) Human Connectome Project (HCP) is to characterize the structural network map of the human brain and identify its associations with covariates such as genotype, risk factors, and so on that correspond to an individual. But the set of image derived measures and the set of covariates are both large, so we must first estimate a 'parsimonious' set of relations between the measurements. For instance, a Gaussian graphical model will show conditional independences between the random variables, which can then be used to setup specific downstream analyses. But most such data involve a large list of 'latent' variables that remain unobserved, yet affect the 'observed' variables sustantially. Accounting for such latent variables is not directly addressed by standard precision matrix estimation, and is tackled via highly specialized optimization methods. This paper offers a unique harmonic analysis view of this problem. By casting the estimation of the precision matrix in terms of a composition of low-frequency latent variables and high-frequency sparse terms, we show how the problem can be formulated using a new wavelet-type expansion in non-Euclidean spaces. Our formulation poses the estimation problem in the frequency space and shows how it can be solved by a simple sub-gradient scheme. We provide a set of scientific results on ~500 scans from the recently released HCP data where our algorithm recovers highly interpretable and sparse conditional dependencies between brain connectivity pathways and well-known covariates.

Probability density functions (PDFs) are fundamental objects in mathematics with numerous applications in computer vision, machine learning and medical imaging. The feasibility of basic operations such as computing the distance between two PDFs and estimating a mean of a set of PDFs is a direct function of the representation we choose to work with. In this paper, we study the Gaussian mixture model (GMM) representation of the PDFs motivated by its numerous attractive features. (1) GMMs are arguably more interpretable than, say, square root parameterizations (2) the model complexity can be explicitly controlled by the number of components and (3) they are already widely used in many applications. The main contributions of this paper are numerical algorithms to enable basic operations on such objects that strictly respect their underlying geometry. For instance, when operating with a set of K component GMMs, a first order expectation is that the result of simple operations like interpolation and averaging should provide an object that is also a K component GMM. The literature provides very little guidance on enforcing such requirements systematically. It turns out that these tasks are important internal modules for analysis and processing of a field of ensemble average propagators (EAPs), common in diffusion weighted magnetic resonance imaging. We provide proof of principle experiments showing how the proposed algorithms for interpolation can facilitate statistical analysis of such data, essential to many neuroimaging studies. Separately, we also derive interesting connections of our algorithm with functional spaces of Gaussians, that may be of independent interest.

Statistical models for manifold-valued data permit capturing the intrinsic nature of the curved spaces in which the data lie and have been a topic of research for several decades. Typically, these formulations use geodesic curves and distances defined locally for most cases - this makes it hard to design parametric models globally on smooth manifolds. Thus, most (manifold specific) parametric models available today assume that the data lie in a small neighborhood on the manifold. To address this 'locality' problem, we propose a novel nonparametric model which unifies multivariate general linear models (MGLMs) using multiple tangent spaces. Our framework generalizes existing work on (both Euclidean and non-Euclidean) general linear models providing a recipe to globally extend the locally-defined parametric models (using a mixture of local models). By grouping observations into sub-populations at multiple tangent spaces, our method provides insights into the hidden structure (geodesic relationships) in the data. This yields a framework to group observations and discover geodesic relationships between covariates X and manifold-valued responses Y , which we call Dirichlet process mixtures of multivariate general linear models (DP-MGLM) on Riemannian manifolds. Finally, we present proof of concept experiments to validate our model.

Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., ( x i , y i ) pairs, are Euclidean is well studied. The setting where y i is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, x i ∈ R , against a manifold-valued variable, y i ∈ . We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -a manifold-valued dependent variable against multiple independent variables, i.e., f : R n → . Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL ( n )/ O ( n ) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.