Positron emission tomography (PET) imaging is an imaging modality for diagnosing a number of neurological diseases. In contrast to Magnetic Resonance Imaging (MRI), PET is costly and involves injecting a radioactive substance into the patient. Motivated by developments in modality transfer in vision, we study the generation of certain types of PET images from MRI data. We derive new flow-based generative models which we show perform well in this small sample size regime (much smaller than dataset sizes available in standard vision tasks). Our formulation, DUAL-GLOW, is based on two invertible networks and a relation network that maps the latent spaces to each other. We discuss how given the prior distribution, learning the conditional distribution of PET given the MRI image reduces to obtaining the conditional distribution between the two latent codes w.r.t. the two image types. We also extend our framework to leverage "side" information (or attributes) when available. By controlling the PET generation through "conditioning" on age, our model is also able to capture brain FDG-PET (hypometabolism) changes, as a function of age. We present experiments on the Alzheimers Disease Neuroimaging Initiative (ADNI) dataset with 826 subjects, and obtain good performance in PET image synthesis, qualitatively and quantitatively better than recent works.

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, efficientformulations for parametric models (which may offer benefits in small sample size regimes) have only emerged recently. Sofar, manifold-valued regression models (such as geodesic regression) are restricted to the analysis of crosssectional 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 offixed 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 : Rd→ 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.

The adoption of "human-in-the-loop" paradigms in computer vision and machine learning is leading to various applications where the actual data acquisition (e.g., human supervision) and the underlying inference algorithms are closely interwined. While classical work in active learning provides effective solutions when the learning module involves classification and regression tasks, many practical issues such as partially observed measurements, financial constraints and even additional distributional or structural aspects of the data typically fall outside the scope of this treatment. For instance, with sequential acquisition of partial measurements of data that manifest as a matrix (or tensor), novel strategies for completion (or collaborative filtering) of the remaining entries have only been studied recently. Motivated by vision problems where we seek to annotate a large dataset of images via a crowdsourced platform or alternatively, complement results from a state-of-the-art object detector using human feedback, we study the "completion" problem defined on graphs, where requests for additional measurements must be made sequentially. We design the optimization model in the Fourier domain of the graph describing how ideas based on adaptive submodularity provide algorithms that work well in practice. On a large set of images collected from Imgur, we see promising results on images that are otherwise difficult to categorize. We also show applications to an experimental design problem in neuroimaging.

Many studies in biomedical and health sciences involve small sample sizes due to logistic or financial constraints. Often, identifying weak (but scientifically interesting) associations between a set of predictors and a response necessitates pooling datasets from multiple diverse labs or groups. While there is a rich literature in statistical machine learning to address distributional shifts and inference in multi-site datasets, it is less clear when such pooling is guaranteed to help (and when it does not) – independent of the inference algorithms we use. In this paper, we present a hypothesis test to answer this question, both for classical and high dimensional linear regression. We precisely identify regimes where pooling datasets across multiple sites is sensible, and how such policy decisions can be made via simple checks executable on each site before any data transfer ever happens. With a focus on Alzheimer’s disease studies, we present empirical results showing that in regimes suggested by our analysis, pooling a local dataset with data from an international study improves power.

There is a great deal of interest in using large scale brain imaging studies to understand how brain connectivity evolves over time for an individual and how it varies over different levels/quantiles of cognitive function. To do so, one typically performs so-called tractography procedures on diffusion MR brain images and derives measures of brain connectivity expressed as graphs. The nodes correspond to distinct brain regions and the edges encode the strength of the connection. The scientific interest is in characterizing the evolution of these graphs over time or from healthy individuals to diseased. We pose this important question in terms of the Laplacian of the connectivity graphs derived from various longitudinal or disease time points - quantifying its progression is then expressed in terms of coupling the harmonic bases of a full set of Laplacians. We derive a coupled system of generalized eigenvalue problems (and corresponding numerical optimization schemes) whose solution helps characterize the full life cycle of brain connectivity evolution in a given dataset. Finally, we show a set of results on a diffusion MR imaging dataset of middle aged people at risk for Alzheimer's disease (AD), who are cognitively healthy. In such asymptomatic adults, we find that a framework for characterizing brain connectivity evolution provides the ability to predict cognitive scores for individual subjects, and for estimating the progression of participant's brain connectivity into the future.

Budget constrained optimal design of experiments is a classical problem in statistics. Although the optimal design literature is very mature, few efficient strategies are available when these design problems appear in the context of sparse linear models commonly encountered in high dimensional machine learning and statistics. In this work, we study experimental design for the setting where the underlying regression model is characterized by a \ell_1-regularized linear function. We propose two novel strategies: the first is motivated geometrically whereas the second is algebraic in nature. We obtain tractable algorithms for this problem and also hold for a more general class of sparse linear models. We perform an extensive set of experiments, on benchmarks and a large multi-site neuroscience study, showing that the proposed models are effective in practice. The latter experiment suggests that these ideas may play a small role in informing enrollment strategies for similar scientific studies in the short-to-medium term future.

Consider samples from two different data sources $\{\mathbf{x_s^i}\} \sim P_{\rm source}$ and $\{\mathbf{x_t^i}\} \sim P_{\rm target}$. We only observe their transformed versions $h(\mathbf{x_s^i})$ and $g(\mathbf{x_t^i})$, for some known function class $h(\cdot)$ and $g(\cdot)$. Our goal is to perform a statistical test checking if $P_{\rm source}$ = $P_{\rm target}$ while removing the distortions induced by the transformations. This problem is closely related to concepts underlying numerous domain adaptation algorithms, and in our case, is motivated by the need to combine clinical and imaging based biomarkers from multiple sites and/or batches, where this problem is fairly common and an impediment in the conduct of analyses with much larger sample sizes. We develop a framework that addresses this problem using ideas from hypothesis testing on the transformed measurements, where in the distortions need to be estimated {\it in tandem} with the testing. We derive a simple algorithm and study its convergence and consistency properties in detail, and we also provide lower-bound strategies based on recent work in continuous optimization. On a dataset of individuals at risk for neurological disease, our results are competitive with alternative procedures that are twice as expensive and in some cases operationally infeasible to implement.

Statistical analysis of longitudinal or cross sectional brain imaging data to identify effects of neurodegenerative diseases is a fundamental task in various studies in neuroscience. However, when there are systematic variations in the images due to parameter changes such as changes in the scanner protocol, hardware changes, or when combining data from multi-site studies, the statistical analysis becomes problematic. Motivated by this scenario, the goal of this paper is to develop a unified statistical solution to the problem of systematic variations in statistical image analysis. Based in part on recent literature in harmonic analysis on diffusion maps, we propose an algorithm which compares operators that are resilient to the systematic variations. These operators are derived from the empirical measurements of the image data and provide an efficient surrogate to capturing the actual changes across images. We also establish a connection between our method to the design of wavelets in non-Euclidean space. To evaluate the proposed ideas, we present various experimental results on detecting changes in simulations as well as show how the method offers improved statistical power in the analysis of real longitudinal PIB-PET imaging data acquired from participants at risk for Alzheimer's disease (AD).

Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a "black box" can often become restrictive. Many 'human in the loop' settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other 'views' of the disease pathology, involving clinical measurements and other image-derived representations.

A variety of studies in neuroscience/neuroimaging seek to perform statistical inference on the acquired brain image scans for diagnosis as well as understanding the pathological manifestation of diseases. To do so, an important first step is to register (or co-register) all of the image data into a common coordinate system. This permits meaningful comparison of the intensities at each voxel across groups (e.g., diseased versus healthy) to evaluate the effects of the disease and/or use machine learning algorithms in a subsequent step. But errors in the underlying registration make this problematic, they either decrease the statistical power or make the follow-up inference tasks less effective/accurate. In this paper, we derive a novel algorithm which offers immunity to local errors in the underlying deformation field obtained from registration procedures. By deriving a deformation invariant representation of the image, the downstream analysis can be made more robust as if one had access to a (hypothetical) far superior registration procedure. Our algorithm is based on recent work on scattering transform. Using this as a starting point, we show how results from harmonic analysis (especially, non-Euclidean wavelets) yields strategies for designing deformation and additive noise invariant representations of large 3-D brain image volumes. We present a set of results on synthetic and real brain images where we achieve robust statistical analysis even in the presence of substantial deformation errors; here, standard analysis procedures significantly under-perform and fail to identify the true signal.

Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., (xi, yi) pairs, are Euclidean is well studied. The setting where yi 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, xi∈ R, against a manifold-valued variable, yi∈ M. 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: ℝn→ M. 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.

Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while being simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics. Permutation testing, on the other hand, is an exact, non parametric method of estimating the FWER for a given α threshold, but for acceptably low thresholds the computational burden can be prohibitive. In this paper, we observe that permutation testing in fact amounts to populating the columns of a very large matrix P. By analyzing the spectrum of this matrix, under certain conditions, we see that P has a low-rank plus a low-variance residual decomposition which makes it suitable for highly sub–sampled — on the order of 0.5% — matrix completion methods. Thus, we propose a novel permutation testing methodology which offers a large speedup, without sacrificing the fidelity of the estimated FWER. Our valuations on four different neuroimaging datasets show that a computational speedup factor of roughly 50× can be achieved while recovering the FWER distribution up to very high accuracy. Further, we show that the estimated α threshold is also recovered faithfully, and is stable.

Multiple Kernel Learning (MKL) generalizes SVMs to the setting where one simultaneously trains a linear classifier and chooses an optimal combination of given base kernels. Model complexity is typically controlled using various norm regularizations on the vector of base kernel mixing coefficients. Existing methods, however, neither regularize nor exploit potentially useful information pertaining to how kernels in the input set 'interact'; that is, higher order kernel-pair relationships that can be easily obtained via unsupervised (similarity, geodesics), supervised (correlation in errors), or domain knowledge driven mechanisms (which features were used to construct the kernel?). We show that by substituting the norm penalty with an arbitrary quadratic function Q \succeq 0, one can impose a desired covariance structure on mixing coefficient selection, and use this as an inductive bias when learning the concept. This formulation significantly generalizes the widely used 1- and 2-norm MKL objectives. We explore the model’s utility via experiments on a challenging Neuroimaging problem, where the goal is to predict a subject’s conversion to Alzheimer’s Disease (AD) by exploiting aggregate information from several distinct imaging modalities. Here, our new model outperforms the state of the art (p-values << 10−3 ). We briefly discuss ramifications in terms of learning bounds (Rademacher complexity).

Hypothesis testing on signals defined on surfaces (such as the cortical surface) is a fundamental component of a variety of studies in Neuroscience. The goal here is to identify regions that exhibit changes as a function of the clinical condition under study. As the clinical questions of interest move towards identifying very early signs of diseases, the corresponding statistical differences at the group level invariably become weaker and increasingly hard to identify. Indeed, after a multiple comparisons correction is adopted (to account for correlated statistical tests over all surface points), very few regions may survive. In contrast to hypothesis tests on point-wise measurements, in this paper, we make the case for performing statistical analysis on multi-scale shape descriptors that characterize the local topological context of the signal around each surface vertex. Our descriptors are based on recent results from harmonic analysis, that show how wavelet theory extends to non-Euclidean settings (i.e., irregular weighted graphs). We provide strong evidence that these descriptors successfully pick up group-wise differences, where traditional methods either fail or yield unsatisfactory results. Other than this primary application, we show how the framework allows performing cortical surface smoothing in the native space without mappint to a unit sphere.

Alzheimer's disease (AD) research has recently witnessed a great deal of activity focused on developing new statistical learning tools for automated inference using imaging data. The workhorse for many of these techniques is the support vector machine (SVM) framework (or more generally kernel-based methods). Most of these require, as a first step, specification of a kernel matrix K between input examples (i.e., images). The inner product between images I(i) and I(j) in a feature space can generally be written in closed form and so it is convenient to treat K as "given." However, in certain neuroimaging applications such an assumption becomes problematic. As an example, it is rather challenging to provide a scalar measure of similarity between two instances of highly attributed data such as cortical thickness measures on cortical surfaces. Note that cortical thickness is known to be discriminative for neurological disorders, so leveraging such information in an inference framework, especially within a multi-modal method, is potentially advantageous. But despite being clinically meaningful, relatively few works have successfully exploited this measure for classification or regression. Motivated by these applications, our paper presents novel techniques to compute similarity matrices for such topologically-based attributed data. Our ideas leverage recent developments to characterize signals (e.g., cortical thickness) motivated by the persistence of their topological features, leading to a scheme for simple constructions of kernel matrices. As a proof of principle, on a dataset of 356 subjects from the Alzheimer's Disease Neuroimaging Initiative study, we report good performance on several statistical inference tasks without any feature selection, dimensionality reduction, or parameter tuning.

Graph-cuts based algorithms are effective for a variety of segmentation tasks in computer vision. Ongoing research is focused toward making the algorithms even more general, as well as to better understand their behavior with respect to issues such as the choice of the weighting function and sensitivity to placement of seeds. In this paper, we investigate in the context of neuroimaging segmentation, the sensitivity/stability of the solution with respect to the input “labels” or “seeds”. In particular, as a form of parameter learning, we are interested in the effect of allowing the given set of labels (and consequently, the response/statistics of the weighting function) to vary for obtaining lower energy segmentation solutions. This perturbation leads to a “refined” label set (or parameters) better suited to the input image, yielding segmentations that are less sensitive to the set of labels or seeds provided. Our proposed algorithm (using Parametric Pseudoflow) yields improvements over graph-cuts based segmentation with a fixed set of labels. We present experiments on about 450 3-D brain image volumes demonstrating the efficacy of the algorithm.

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