In this paper we refine the process of computing calibration functions for a number of multiclass classification surrogate losses. Calibration functions are a powerful tool for easily converting bounds for the surrogate risk (which can be computed through well-known methods) into bounds for the true risk, the probability of making a mistake. They are particularly suitable in non-parametric settings, where the approximation error can be controlled, and provide tighter bounds than the common technique of upper-bounding the 0-1 loss by the surrogate loss. The abstract nature of the more sophisticated existing calibration function results requires calibration functions to be explicitly derived on a case-by-case basis, requiring repeated efforts whenever bounds for a new surrogate loss are required. We devise a streamlined analysis that simplifies the process of deriving calibration functions for a large number of surrogate losses that have been proposed in the literature. The effort of deriving calibration functions is then surmised in verifying, for a chosen surrogate loss, a small number of conditions that we introduce. As case studies, we recover existing calibration functions for the well-known loss of Lee et al. (2004), and also provide novel calibration functions for well-known losses, including the one-versus-all loss and the logistic regression loss, plus a number of other losses that have been shown to be classification-calibrated in the past, but for which no calibration function had been derived.

Sparse versions of principal component analysis (PCA) have imposed themselves as simple, yet powerful ways of selecting relevant features of high-dimensional data in an unsupervised manner. However, when several sparse principal components are computed, the interpretation of the selected variables is difficult since each axis has its own sparsity pattern and has to be interpreted separately. To overcome this drawback, we propose a Bayesian procedure called globally sparse probabilistic PCA (GSPPCA) that allows to obtain several sparse components with the same sparsity pattern. This allows the practitioner to identify the original variables which are relevant to describe the data. To this end, using Roweis' probabilistic interpretation of PCA and a Gaussian prior on the loading matrix, we provide the first exact computation of the marginal likelihood of a Bayesian PCA model. To avoid the drawbacks of discrete model selection, a simple relaxation of this framework is presented. It allows to find a path of models using a variational expectation-maximization algorithm. The exact marginal likelihood is then maximized over this path. This approach is illustrated on real and synthetic data sets. In particular, using unlabeled microarray data, GSPPCA infers much more relevant gene subsets than traditional sparse PCA algorithms.

The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that it requires solving a non-convex optimization problem. This paper shows a remarkable result: despite the non-convexity of the TREX problem, there exists a polynomial-time algorithm that is guaranteed to find the global minimum. This result adds the TREX to a very short list of non-convex optimization problems that can be globally optimized (principal components analysis being a famous example). After deriving and developing this new approach, we demonstrate that (i) the ability of the preexisting TREX heuristic to reach the global minimum is strongly dependent on the difficulty of the underlying statistical problem, (ii) the new polynomial-time algorithm for TREX permits a novel variable ranking and selection scheme, (iii) this scheme can be incorporated into a rule that controls the false discovery rate (FDR) of included features in the model. To achieve this last aim, we provide an extension of the results of Barber & Candes (2015) to establish that the knockoff filter framework can be applied to the TREX. This investigation thus provides both a rare case study of a heuristic for non-convex optimization and a novel way of exploiting non-convexity for statistical inference.

This work proposes a new algorithm for automated and simultaneous phenotyping of multiple co-occurring medical conditions, also referred as comorbidities, using clinical notes from the electronic health records (EHRs). A basic latent factor estimation technique of non-negative matrix factorization (NMF) is augmented with domain specific constraints to obtain sparse latent factors that are anchored to a fixed set of chronic conditions. The proposed anchoring mechanism ensures a one-to-one identifiable and interpretable mapping between the latent factors and the target comorbidities. Qualitative assessment of the empirical results by clinical experts suggests that the proposed model learns clinically interpretable phenotypes while being predictive of 30 day mortality. The proposed method can be readily adapted to any non-negative EHR data across various healthcare institutions.

The goal of this paper is to show that there exists a simple, yet universal statistical logic of spectral graph analysis by recasting it into a nonparametric function estimation problem. The prescribed viewpoint appears to be good enough to accommodate most of the existing spectral graph techniques as a consequence of just one single formalism and algorithm.

The goal of this thesis is to investigate the potential of predictive modelling for football injuries. This work was conducted in close collaboration with Tottenham Hotspurs FC (THFC), the PGA European tour and the participation of Wolverhampton Wanderers (WW). Three investigations were conducted: 1. Predicting the recovery time of football injuries using the UEFA injury recordings: The UEFA recordings is a common standard for recording injuries in professional football. For this investigation, three datasets of UEFA injury recordings were available. Different machine learning algorithms were used in order to build a predictive model. The performance of the machine learning models is then improved by using feature selection conducted through correlation-based subset feature selection and random forests. 2. Predicting injuries in professional football using exposure records: The relationship between exposure (in training hours and match hours) in professional football athletes and injury incidence was studied. A common problem in football is understanding how the training schedule of an athlete can affect the chance of him getting injured. The task was to predict the number of days a player can train before he gets injured. 3. Predicting intrinsic injury incidence using in-training GPS measurements: A significant percentage of football injuries can be attributed to overtraining and fatigue. GPS data collected during training sessions might provide indicators of fatigue, or might be used to detect very intense training sessions which can lead to overtraining. This research used GPS data gathered during training sessions of the first team of THFC, in order to predict whether an injury would take place during a week.

BigML's Summer 2016 Release is here! GMT 02:00) for a FREE live webinar to learn about the newest version of BigML. We'll be diving into Logistic Regression, one of the most popular supervised Machine Learning methods for solving classification problems. Last Fall we launched Logistic Regressions in the BigML API to let you easily create and download models to your environment for fast, local predictions. With this Summer Release, we go a step further by bringing Logistic Regression to the BigML Dashboard.

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.

General predictive models do not provide a measure of confidence in predictions without Bayesian assumptions. A way to circumvent potential restrictions is to use conformal methods for constructing non-parametric confidence regions, that offer guarantees regarding validity. In this paper we provide a detailed description of a computationally efficient conformal procedure for Kernel Ridge Regression (KRR), and conduct a comparative numerical study to see how well conformal regions perform against the Bayesian confidence sets. The results suggest that conformalized KRR can yield predictive confidence regions with specified coverage rate, which is essential in constructing anomaly detection systems based on predictive models.

Principal component analysis (PCA) is often used to reduce the dimension of data by selecting a few orthonormal vectors that explain most of the variance structure of the data. L1 PCA uses the L1 norm to measure error, whereas the conventional PCA uses the L2 norm. For the L1 PCA problem minimizing the fitting error of the reconstructed data, we propose an exact reweighted and an approximate algorithm based on iteratively reweighted least squares. We provide convergence analyses, and compare their performance against benchmark algorithms in the literature. The computational experiment shows that the proposed algorithms consistently perform best.