Over the course of the past decade, a variety of randomized algorithms have been proposed for computing approximate least-squares (LS) solutions in large-scale settings. A longstanding practical issue is that, for any given input, the user rarely knows the actual error of an approximate solution (relative to the exact solution). Likewise, it is difficult for the user to know precisely how much computation is needed to achieve the desired error tolerance. Consequently, the user often appeals to worst-case error bounds that tend to offer only qualitative guidance. As a more practical alternative, we propose a bootstrap method to compute a posteriori error estimates for randomized LS algorithms. These estimates permit the user to numerically assess the error of a given solution, and to predict how much work is needed to improve a "preliminary" solution. In addition, we provide theoretical consistency results for the method, which are the first such results in this context (to the best of our knowledge). From a practical standpoint, the method also has considerable flexibility, insofar as it can be applied to several popular sketching algorithms, as well as a variety of error metrics. Moreover, the extra step of error estimation does not add much cost to an underlying sketching algorithm. Finally, we demonstrate the effectiveness of the method with empirical results.

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary supervision data, these directions continuously bridge the principal component, mean difference and linear discriminant directions, thus ranging from unsupervised to fully supervised dimension reduction. High-dimensional asymptotic studies of continuum directions for binary supervision reveal several interesting facts. The conditions under which the sample continuum directions are inconsistent, but their classification performance is good, are specified. While the proposed method can be directly used for binary and multi-category classification, its generalizations to incorporate any form of auxiliary data are also presented. The proposed method enjoys fast computation, and the performance is better or on par with more computer-intensive alternatives. Keywords: continuum regression, dimension reduction, linear discriminant analysis, high-dimension, low-sample-size (HDLSS), maximum data piling, principal component analysis 2000 MSC: 60K35 1. Introduction In modern complex data, it becomes increasingly common that multiple data sets are available. Two types of data are collected on a same set of subjects: a data set of primary interestX and an auxiliary data setY . The goal of supervised dimension reduction is to delineate major signals inX, dependent toY . Relevant application areas include genomics (genetic studies collect both gene expression and SNP data--Li et al. (2016)), finance data (stocks asX in relation to characteristicsY of each stock: size, value, momentum and volatility--Connor et al. (2012)), and batch effect adjustments (Lee et al., 2014). There has been a number of work in dealing with the multi-source data situation. Lock et al. (2013) developed JIVE to separate joint variation from individual variations. Large-scale correlation studies can identify millions of pairwise associations between two data sets via multiple canonical correlation analysis (Witten and Tibshirani, 2009). These methods, however, do not provide supervised dimension reduction of a particular data setX, since all data sets assume an equal role. In contrast, reduced-rank regression (RRR, Izenman, 1975; Tso, 1981) and envelop models (Cook et al., 2010) provide sufficient dimension reduction (Cook and Ni, 2005) for regression problems. See Cook et al. (2013) for connections between envelops and partial least square regression.

This work introduces a novel estimation method, called LOVE, of the entries and structure of a loading matrix A in a sparse latent factor model X = AZ + E, for an observable random vector X in Rp, with correlated unobservable factors Z \in RK, with K unknown, and independent noise E. Each row of A is scaled and sparse. In order to identify the loading matrix A, we require the existence of pure variables, which are components of X that are associated, via A, with one and only one latent factor. Despite the fact that the number of factors K, the number of the pure variables, and their location are all unknown, we only require a mild condition on the covariance matrix of Z, and a minimum of only two pure variables per latent factor to show that A is uniquely defined, up to signed permutations. Our proofs for model identifiability are constructive, and lead to our novel estimation method of the number of factors and of the set of pure variables, from a sample of size n of observations on X. This is the first step of our LOVE algorithm, which is optimization-free, and has low computational complexity of order p2. The second step of LOVE is an easily implementable linear program that estimates A. We prove that the resulting estimator is minimax rate optimal up to logarithmic factors in p. The model structure is motivated by the problem of overlapping variable clustering, ubiquitous in data science. We define the population level clusters as groups of those components of X that are associated, via the sparse matrix A, with the same unobservable latent factor, and multi-factor association is allowed. Clusters are respectively anchored by the pure variables, and form overlapping sub-groups of the p-dimensional random vector X. The Latent model approach to OVErlapping clustering is reflected in the name of our algorithm, LOVE.

Determining how certain properties are related to other properties is fundamental to scientific discovery. As data collection rates accelerate, it is becoming increasingly difficult yet ever more important to determine whether one property of data (e.g., cloud density) is related to another (e.g., grass wetness). Only if two properties are related are further investigations into the geometry of the relationship warranted. While existing approaches can test whether two properties are related, they may require unfeasibly large sample sizes in real data scenarios, and do not address how they are related. Our key insight is that one can adaptively restrict the analysis to the "jointly local" observations---that is, one can estimate the scales with the most informative neighbors for determining the existence and geometry of a relationship. "Multiscale Graph Correlation" (MGC) is a framework that extends global procedures to be multiscale; consequently, MGC tests typically require far fewer samples than existing methods for a wide variety of dependence structures and dimensionalities, while maintaining computational efficiency. Moreover, MGC provides a simple and elegant multiscale characterization of the potentially complex latent geometry underlying the relationship. In several real data applications, MGC uniquely detects the presence and reveals the geometry of the relationships.

Generative adversarial networks (GAN) are a powerful subclass of generative models. Despite a very rich research activity leading to numerous interesting GAN algorithms, it is still very hard to assess which algorithm(s) perform better than others. We conduct a neutral, multi-faceted large-scale empirical study on state-of-the art models and evaluation measures. We find that most models can reach similar scores with enough hyperparameter optimization and random restarts. This suggests that improvements can arise from a higher computational budget and tuning more than fundamental algorithmic changes. To overcome some limitations of the current metrics, we also propose several data sets on which precision and recall can be computed. Our experimental results suggest that future GAN research should be based on more systematic and objective evaluation procedures. Finally, we did not find evidence that any of the tested algorithms consistently outperforms the original one.

The central aim in this paper is to address variable selection questions in nonlinear and nonparametric regression. Motivated by statistical genetics, where nonlinear interactions are of particular interest, we introduce a novel, interpretable, and computationally efficient way to summarize the relative importance of predictor variables. Methodologically, we develop the "RelATive cEntrality" (RATE) measure to prioritize candidate genetic variants that are not just marginally important, but whose associations also stem from significant covarying relationships with other variants in the data. We illustrate RATE through Bayesian Gaussian process regression, but the methodological innovations apply to other nonlinear methods. It is known that nonlinear models often exhibit greater predictive accuracy than linear models, particularly for phenotypes generated by complex genetic architectures. With detailed simulations and an Arabidopsis thaliana QTL mapping study, we show that applying RATE enables an explanation for this improved performance.

Magnetic resonance imaging (MRI) has been proposed as a complimentary method to measure bone quality and assess fracture risk. However, manual segmentation of MR images of bone is time-consuming, limiting the use of MRI measurements in the clinical practice. The purpose of this paper is to present an automatic proximal femur segmentation method that is based on deep convolutional neural networks (CNNs). This study had institutional review board approval and written informed consent was obtained from all subjects. A dataset of volumetric structural MR images of the proximal femur from 86 subject were manually-segmented by an expert. We performed experiments by training two different CNN architectures with multiple number of initial feature maps and layers, and tested their segmentation performance against the gold standard of manual segmentations using fourfold cross-validation. Automatic segmentation of the proximal femur achieved a high dice similarity score of 0.94 0.05 with precision 0.95 0.02, and recall 0.94 0.08 using a CNN architecture based on 3D convolution exceeding the performance of 2D CNNs. The high segmentation accuracy provided by CNNs has the potential to help bring the use of structural MRI measurements of bone quality into clinical practice for management of osteoporosis.

In order to reach human performance on complex visual tasks, artificial systems need to incorporate a significant amount of understanding of the world in terms of macroscopic objects, movements, forces, etc. Inspired by work on intuitive physics in infants, we propose an evaluation framework which diagnoses how much a given system understands about physics by testing whether it can tell apart well matched videos of possible versus impossible events. The test requires systems to compute a physical plausibility score over an entire video. It is free of bias and can test a range of specific physical reasoning skills. We then describe the first release of a benchmark dataset aimed at learning intuitive physics in an unsupervised way, using videos constructed with a game engine. We describe two Deep Neural Network baseline systems trained with a future frame prediction objective and tested on the possible versus impossible discrimination task. The analysis of their results compared to human data gives novel insights in the potentials and limitations of next frame prediction architectures.

Estimating multiple sparse Gaussian Graphical Models (sGGMs) jointly for many related tasks (large $K$) under a high-dimensional (large $p$) situation is an important task. Most previous studies for the joint estimation of multiple sGGMs rely on penalized log-likelihood estimators that involve expensive and difficult non-smooth optimizations. We propose a novel approach, FASJEM for \underline{fa}st and \underline{s}calable \underline{j}oint structure-\underline{e}stimation of \underline{m}ultiple sGGMs at a large scale. As the first study of joint sGGM using the Elementary Estimator framework, our work has three major contributions: (1) We solve FASJEM through an entry-wise manner which is parallelizable. (2) We choose a proximal algorithm to optimize FASJEM. This improves the computational efficiency from $O(Kp^3)$ to $O(Kp^2)$ and reduces the memory requirement from $O(Kp^2)$ to $O(K)$. (3) We theoretically prove that FASJEM achieves a consistent estimation with a convergence rate of $O(\log(Kp)/n_{tot})$. On several synthetic and four real-world datasets, FASJEM shows significant improvements over baselines on accuracy, computational complexity, and memory costs.

Lung cancer is the leading cause of cancer-related death worldwide. Early diagnosis of pulmonary nodules in Computed Tomography (CT) chest scans provides an opportunity for designing effective treatment and making financial and care plans. In this paper, we consider the problem of diagnostic classification between benign and malignant lung nodules in CT images, which aims to learn a direct mapping from 3D images to class labels. To achieve this goal, four two-pathway Convolutional Neural Networks (CNN) are proposed, including a basic 3D CNN, a novel multi-output network, a 3D DenseNet, and an augmented 3D DenseNet with multi-outputs. These four networks are evaluated on the public LIDC-IDRI dataset and outperform most existing methods. In particular, the 3D multi-output DenseNet (MoDenseNet) achieves the state-of-the-art classification accuracy on the task of end-to-end lung nodule diagnosis. In addition, the networks pretrained on the LIDC-IDRI dataset can be further extended to handle smaller datasets using transfer learning. This is demonstrated on our dataset with encouraging prediction accuracy in lung nodule classification.