Research into disease-related health variables typically involve choosing health variables and conditions, and using statistical methods to study the strength of association of the variables with the condition [9]. These are then used to confirm known or suspected relationships between the behavioural/health factors or disease conditions. There may be information about health status that may be gleaned by considering different aspects of an individual's data, and investigating possible relationships between the variables. Representations that capture these relationships can be useful in predicting presence or risk level of medical conditions. The National Health and Nutrition Examination Survey (NHANES) dataset provides data on health measurements, taken from survey participants, comprising different categories including demographics, laboratory tests and physical measurements.

Explicit high-order feature interactions efficiently capture essential structural knowledge about the data of interest and have been used for constructing generative models. We present a supervised discriminative High-Order Parametric Embedding (HOPE) approach to data visualization and compression. Compared to deep embedding models with complicated deep architectures, HOPE generates more effective high-order feature mapping through an embarrassingly simple shallow model. Furthermore, two approaches to generating a small number of exemplars conveying high-order interactions to represent large-scale data sets are proposed. These exemplars in combination with the feature mapping learned by HOPE effectively capture essential data variations. Moreover, through HOPE, these exemplars are employed to increase the computational efficiency of kNN classification for fast information retrieval by thousands of times. For classification in two-dimensional embedding space on MNIST and USPS datasets, our shallow method HOPE with simple Sigmoid transformations significantly outperforms state-of-the-art supervised deep embedding models based on deep neural networks, and even achieved historically low test error rate of 0.65% in two-dimensional space on MNIST, which demonstrates the representational efficiency and power of supervised shallow models with high-order feature interactions.

Abstract--We consider N -way data arrays and low-rank tensor factorizations where the time mode is coded as a sparse linear combination of temporal elements from an over-complete library. Our method, Shape Constrained T ensor Decomposition (SCTD) is based upon the CANDECOMP/PARAF AC (CP) decomposition which produces r -rank approximations of data tensors via outer products of vectors in each dimension of the data. By constraining the vector in the temporal dimension to known analytic forms which are selected from a large set of candidate functions, more readily interpretable decompositions are achieved and analytic time dependencies discovered. The SCTD method circumvents traditional flattening techniques where an N -way array is reshaped into a matrix in order to perform a singular value decomposition. A clear advantage of the SCTD algorithm is its ability to extract transient and intermittent phenomena which is often difficult for SVD-based methods. We motivate the SCTD method using several intuitively appealing results before applying it on a number of high-dimensional, real-world data sets in order to illustrate the efficiency of the algorithm in extracting interpretable spatiotemporal modes. With the rise of data-driven discovery methods, the decomposition proposed provides a viable technique for analyzing multitudes of data in a more comprehensible fashion. A TRIX decompositions are critically enabling algorithms for scientific computing and data analysis applications across every field of the engineering, social, biological, and physical sciences. Of particular importance is the singular value decomposition (SVD), which provides a principled method for dimensionality reduction and computation of interpretable subspaces within which the data reside. So widespread is the usage of the algorithm, and minor modifications thereof, that it has generated a myriad of names across various communities, including Principal Component Analysis (PCA) [1], the Karhunen-Lo eve (KL) decomposition, Hotelling transform [2], [3], Empirical Orthogonal Functions (EOFs) [4] and Proper Orthogonal Decomposition (POD) [5], [6]. However, in order to use the SVD, data, which generally may be of N distinct dimensions, must be flattened into a matrix form, potentially compromising the statistical accuracy of the subspaces computed. B. Lusch and J. N. Kutz are with the Department of Applied Mathematics, University of Washington, Seattle, W A 98195-3925 USA email: herwaldt@uw.edu, E. Chi is with the Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203 USA email: eric chi@ncsu.edu. It is often the case that one of the dimensions considered in the tensor is the time variable.

Prediction of the future trajectory of a disease is an important challenge for personalized medicine and population health management. However, many complex chronic diseases exhibit large degrees of heterogeneity, and furthermore there is not always a single readily available biomarker to quantify disease severity. Even when such a clinical variable exists, there are often additional related biomarkers routinely measured for patients that may better inform the predictions of their future disease state. To this end, we propose a novel probabilistic generative model for multivariate longitudinal data that captures dependencies between multivariate trajectories. We use a Gaussian process based regression model for each individual trajectory, and build off ideas from latent class models to induce dependence between their mean functions. We fit our method using a scalable variational inference algorithm to a large dataset of longitudinal electronic patient health records, and find that it improves dynamic predictions compared to a recent state of the art method. Our local accountable care organization then uses the model predictions during chart reviews of high risk patients with chronic kidney disease.

In this paper, we propose a novel learning framework for the problem of domain transfer learning. We map the data of two domains to one single common space, and learn a classifier in this common space. Then we adapt the common classifier to the two domains by adding two adaptive functions to it respectively. In the common space, the target domain data points are weighted and matched to the target domain in term of distributions. The weighting terms of source domain data points and the target domain classification responses are also regularized by the local reconstruction coefficients. The novel transfer learning framework is evaluated over some benchmark cross-domain data sets, and it outperforms the existing state-of-the-art transfer learning methods.

A novel efficient method for computing the Knowledge-Gradient policy for Continuous Parameters (KGCP) for deterministic optimization is derived. The differences with Expected Improvement (EI), a popular choice for Bayesian optimization of deterministic engineering simulations, are explored. Both policies and the Upper Confidence Bound (UCB) policy are compared on a number of benchmark functions including a problem from structural dynamics. It is empirically shown that KGCP has similar performance as the EI policy for many problems, but has better convergence properties for complex (multi-modal) optimization problems as it emphasizes more on exploration when the model is confident about the shape of optimal regions. In addition, the relationship between Maximum Likelihood Estimation (MLE) and slice sampling for estimation of the hyperparameters of the underlying models, and the complexity of the problem at hand, is studied.

These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.

In the probabilistic topic models, the quantity of interest---a low-rank matrix consisting of topic vectors---is hidden in the text corpus matrix, masked by noise, and the Singular Value Decomposition (SVD) is a potentially useful tool for learning such a low-rank matrix. However, the connection between this low-rank matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges. In this paper, we overcome the challenge by revealing a surprising insight: there is a low-dimensional simplex structure which can be viewed as a bridge between the low-rank matrix of interest and the SVD of the text corpus matrix, and allows us to conveniently reconstruct the former using the latter. Such an insight motivates a new SVD approach to learning topic models, which we analyze with delicate random matrix theory and derive the rate of convergence. We support our methods and theory numerically, using both simulated data and real data.

When working with three-dimensional data, choice of representation is key. We explore voxel-based models, and present evidence for the viability of voxellated representations in applications including shape modeling and object classification. Our key contributions are methods for training voxel-based variational autoencoders, a user interface for exploring the latent space learned by the autoencoder, and a deep convolutional neural network architecture for object classification. We address challenges unique to voxel-based representations, and empirically evaluate our models on the ModelNet benchmark, where we demonstrate a 51.5% relative improvement in the state of the art for object classification.

Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.