Tensor-valued data arise frequently from a wide variety of scientific applications, and many among them can be translated into an alteration detection problem of tensor dependence structures. In this article, we formulate the problem under the popularly adopted tensor-normal distributions and aim at two-sample correlation/partial correlation comparisons of tensor-valued observations. Through decorrelation and centralization, a separable covariance structure is employed to pool sample information from different tensor modes to enhance the power of the test. Additionally, we propose a novel Sparsity-Exploited Reranking Algorithm (SERA) to further improve the multiple testing efficiency. The algorithm is approached through reranking of the p-values derived from the primary test statistics, by incorporating a carefully constructed auxiliary tensor sequence. Besides the tensor framework, SERA is also generally applicable to a wide range of two-sample large-scale inference problems with sparsity structures, and is of independent interest. The asymptotic properties of the proposed test are derived and the algorithm is shown to control the false discovery at the pre-specified level. We demonstrate the efficacy of the proposed method through intensive simulations and two scientific applications.

Although Alzheimer's disease (AD) cannot be reversed or cured, timely diagnosis can significantly reduce the burden of treatment and care. Current research on AD diagnosis models usually regards the diagnosis task as a typical classification task with two primary assumptions: 1) All target categories are known a priori; 2) The diagnostic strategy for each patient is consistent, that is, the number and type of model input data for each patient are the same. However, real-world clinical settings are open, with complexity and uncertainty in terms of both subjects and the resources of the medical institutions. This means that diagnostic models may encounter unseen disease categories and need to dynamically develop diagnostic strategies based on the subject's specific circumstances and available medical resources. Thus, the AD diagnosis task is tangled and coupled with the diagnosis strategy formulation. To promote the application of diagnostic systems in real-world clinical settings, we propose OpenClinicalAI for direct AD diagnosis in complex and uncertain clinical settings. This is the first powerful end-to-end model to dynamically formulate diagnostic strategies and provide diagnostic results based on the subject's conditions and available medical resources. OpenClinicalAI combines reciprocally coupled deep multiaction reinforcement learning (DMARL) for diagnostic strategy formulation and multicenter meta-learning (MCML) for open-set recognition. The experimental results show that OpenClinicalAI achieves better performance and fewer clinical examinations than the state-of-the-art model. Our method provides an opportunity to embed the AD diagnostic system into the current health care system to cooperate with clinicians to improve current health care.

Alzheimer's disease (AD) cannot be reversed, but early diagnosis will significantly benefit patients' medical treatment and care. In recent works, AD diagnosis has the primary assumption that all categories are known a prior -- a closed-set classification problem, which contrasts with the open-set recognition problem. This assumption hinders the application of the model in natural clinical settings. Although many open-set recognition technologies have been proposed in other fields, they are challenging to use for AD diagnosis directly since 1) AD is a degenerative disease of the nervous system with similar symptoms at each stage, and it is difficult to distinguish from its pre-state, and 2) diversified strategies for AD diagnosis are challenging to model uniformly. In this work, inspired by the concerns of clinicians during diagnosis, we propose an open-set recognition model, OpenAPMax, based on the anomaly pattern to address AD diagnosis in real-world settings. OpenAPMax first obtains the abnormal pattern of each patient relative to each known category through statistics or a literature search, clusters the patients' abnormal pattern, and finally, uses extreme value theory (EVT) to model the distance between each patient's abnormal pattern and the center of their category and modify the classification probability. We evaluate the performance of the proposed method with recent open-set recognition, where we obtain state-of-the-art results.

Mixed-membership (MM) models such as Latent Dirichlet Allocation (LDA) have been applied to microbiome compositional data to identify latent subcommunities of microbial species. However, microbiome compositional data, especially those collected from the gut, typically display substantial cross-sample heterogeneities in the subcommunity composition which current MM methods do not account for. To address this limitation, we incorporate the logistic-tree normal (LTN) model -- using the phylogenetic tree structure -- into the LDA model to form a new MM model. This model allows variation in the composition of each subcommunity around some ``centroid'' composition. Incorporation of auxiliary P\'olya-Gamma variables enables a computationally efficient collapsed blocked Gibbs sampler to carry out Bayesian inference under this model. We compare the new model and LDA and show that in the presence of large cross-sample heterogeneity, under the LDA model the resulting inference can be extremely sensitive to the specification of the total number of subcommunities as it does not account for cross-sample heterogeneity. As such, the popular strategy in other applications of MM models of overspecifying the number of subcommunities -- and hoping that some meaningful subcommunities will emerge among artificial ones -- can lead to highly misleading conclusions in the microbiome context. In contrast, by accounting for such heterogeneity, our MM model restores the robustness of the inference in the specification of the number of subcommunities and again allows meaningful subcommunities to be identified under this strategy.

This paper quantitatively reveals the state-of-the-art and state-of-the-practice AI systems only achieve acceptable performance on the stringent conditions that all categories of subjects are known, which we call closed clinical settings, but fail to work in real-world clinical settings. Compared to the diagnosis task in the closed setting, real-world clinical settings pose severe challenges, and we must treat them differently. We build a clinical AI benchmark named Clinical AIBench to set up real-world clinical settings to facilitate researches. We propose an open, dynamic machine learning framework and develop an AI system named OpenClinicalAI to diagnose diseases in real-world clinical settings. The first versions of Clinical AIBench and OpenClinicalAI target Alzheimer's disease. In the real-world clinical setting, OpenClinicalAI significantly outperforms the state-of-the-art AI system. In addition, OpenClinicalAI develops personalized diagnosis strategies to avoid unnecessary testing and seamlessly collaborates with clinicians. It is promising to be embedded in the current medical systems to improve medical services.

Learning probability measures based on an i.i.d. sample is a fundamental inference task, but is challenging when the sample space is high-dimensional. Inspired by the success of tree boosting in high-dimensional classification and regression, we propose a tree boosting method for learning high-dimensional probability distributions. We formulate concepts of "addition'' and "residuals'' on probability distributions in terms of compositions of a new, more general notion of multivariate cumulative distribution functions (CDFs) than classical CDFs. This then gives rise to a simple boosting algorithm based on forward-stagewise (FS) fitting of an additive ensemble of measures. The output of the FS algorithm allows analytic computation of the probability density function for the fitted distribution. It also provides an exact simulator for drawing independent Monte Carlo samples from the fitted measure. Typical considerations in applying boosting -- namely choosing the number of trees, setting the appropriate level of shrinkage/regularization in the weak learner, and the evaluation of variable importance -- can be accomplished in an analogous fashion to traditional boosting in supervised learning. Numerical experiments confirm that boosting can substantially improve the fit to multivariate distributions compared to the state-of-the-art single-tree learner and is computationally efficient. We illustrate through an application to a data set from mass cytometry how the simulator can be used to investigate various aspects of the underlying distribution.

We present a framework for testing independence between two random vectors that is scalable to massive data. Taking a "divide-and-conquer" approach, we break down the nonparametric multivariate test of independence into simple univariate independence tests on a collection of $2\times 2$ contingency tables, constructed by sequentially discretizing the original sample space at a cascade of scales from coarse to fine. This transforms a complex nonparametric testing problem---that traditionally requires quadratic computational complexity with respect to the sample size---into a multiple testing problem that can be addressed with a computational complexity that scales almost linearly with the sample size. We further consider the scenario when the dimensionality of the two random vectors also grows large, in which case the curse of dimensionality arises in the proposed framework through an explosion in the number of univariate tests to be completed. To overcome this difficulty, we propose a data-adaptive version of our method that completes a fraction of the univariate tests, judged to be more likely to contain evidence for dependency based on exploiting the spatial characteristics of the dependency structure in the data. We provide an inference recipe based on multiple testing adjustment that guarantees the inferential validity in terms of properly controlling the family-wise error rate. We demonstrate the tremendous computational advantage of the algorithm in comparison to existing approaches while achieving desirable statistical power through an extensive simulation study. In addition, we illustrate how our method can be used for learning the nature of the underlying dependency in addition to hypothesis testing. We demonstrate the use of our method through analyzing a data set from flow cytometry.

Traditional statistical wavelet analysis that carries out modeling and inference based on wavelet coefficients under a given, predetermined wavelet transform can quickly lose efficiency in multivariate problems, because such wavelet transforms, which are typically symmetric with respect to the dimensions, cannot adaptively exploit the energy distribution in a problem-specific manner. We introduce a principled probabilistic framework for incorporating such adaptivity---by (i) representing multivariate functions using one-dimensional (1D) wavelet transforms applied to a permuted version of the original function, and (ii) placing a prior on the corresponding permutation, thereby forming a mixture of permuted 1D wavelet transforms. Such a representation can achieve substantially better energy concentration in the wavelet coefficients. In particular, when combined with the Haar basis, we show that exact Bayesian inference under the model can be achieved analytically through a recursive message passing algorithm with a computational complexity that scales linearly with sample size. In addition, we propose a sequential Monte Carlo (SMC) inference algorithm for other wavelet bases using the exact Haar solution as the proposal. We demonstrate that with this framework even simple 1D Haar wavelets can achieve excellent performance in both 2D and 3D image reconstruction via numerical experiments, outperforming state-of-the-art multidimensional wavelet-based methods especially in low signal-to-noise ratio settings, at a fraction of the computational cost.

There has been great interest recently in applying nonparametric kernel mixtures in a hierarchical manner to model multiple related data samples jointly. In such settings several data features are commonly present: (i) the related samples often share some, if not all, of the mixture components but with differing weights, (ii) only some, not all, of the mixture components vary across the samples, and (iii) often the shared mixture components across samples are not aligned perfectly in terms of their location and spread, but rather display small misalignments either due to systematic cross-sample difference or more often due to uncontrolled, extraneous causes. Properly incorporating these features in mixture modeling will enhance the efficiency of inference, whereas ignoring them not only reduces efficiency but can jeopardize the validity of the inference due to issues such as confounding. We introduce two techniques for incorporating these features in modeling related data samples using kernel mixtures. The first technique, called $\psi$-stick breaking, is a joint generative process for the mixing weights through the breaking of both a stick shared by all the samples for the components that do not vary in size across samples and an idiosyncratic stick for each sample for those components that do vary in size. The second technique is to imbue random perturbation into the kernels, thereby accounting for cross-sample misalignment. These techniques can be used either separately or together in both parametric and nonparametric kernel mixtures. We derive efficient Bayesian inference recipes based on MCMC sampling for models featuring these techniques, and illustrate their work through both simulated data and a real flow cytometry data set in prediction/estimation, cross-sample calibration, and testing multi-sample differences.

We introduce a wavelet-domain functional analysis of variance (fANOVA) method based on a Bayesian hierarchical model. The factor effects are modeled through a spike-and-slab mixture at each location-scale combination along with a normal-inverse-Gamma (NIG) conjugate setup for the coefficients and errors. A graphical model called the Markov grove (MG) is designed to jointly model the spike-and-slab statuses at all location-scale combinations, which incorporates the clustering of each factor effect in the wavelet-domain thereby allowing borrowing of strength across location and scale. The posterior of this NIG-MG model is analytically available through a pyramid algorithm of the same computational complexity as Mallat's pyramid algorithm for discrete wavelet transform, i.e., linear in both the number of observations and the number of locations. Posterior probabilities of factor contributions can also be computed through pyramid recursion, and exact samples from the posterior can be drawn without MCMC. We investigate the performance of our method through extensive simulation and show that it outperforms existing wavelet-domain fANOVA methods in a variety of common settings. We apply the method to analyzing the orthosis data.