239 research outputs found

    Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

    Full text link
    Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an â„“0\ell_0-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests robust PCA schemes based on convex relaxation, which lead naturally to a family of robust estimators encompassing Huber's optimal M-class as a special case. Outliers are identified by tuning a regularization parameter, which amounts to controlling sparsity of the outlier matrix along the whole robustification path of (group) least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its neat ties to robust statistics, the developed outlier-aware PCA framework is versatile to accommodate novel and scalable algorithms to: i) track the low-rank signal subspace robustly, as new data are acquired in real time; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Synthetic and real data tests corroborate the effectiveness of the proposed robust PCA schemes, when used to identify aberrant responses in personality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data.Comment: 30 pages, submitted to IEEE Transactions on Signal Processin

    Projected Randomized Smoothing for Certified Adversarial Robustness

    Full text link
    Randomized smoothing is the current state-of-the-art method for producing provably robust classifiers. While randomized smoothing typically yields robust â„“2\ell_2-ball certificates, recent research has generalized provable robustness to different norm balls as well as anisotropic regions. This work considers a classifier architecture that first projects onto a low-dimensional approximation of the data manifold and then applies a standard classifier. By performing randomized smoothing in the low-dimensional projected space, we characterize the certified region of our smoothed composite classifier back in the high-dimensional input space and prove a tractable lower bound on its volume. We show experimentally on CIFAR-10 and SVHN that classifiers without the initial projection are vulnerable to perturbations that are normal to the data manifold and yet are captured by the certified regions of our method. We compare the volume of our certified regions against various baselines and show that our method improves on the state-of-the-art by many orders of magnitude.Comment: Transactions on Machine Learning Research (TMLR) 202

    Timescale effect estimation in time-series studies of air pollution and health: A Singular Spectrum Analysis approach

    Full text link
    A wealth of epidemiological data suggests an association between mortality/morbidity from pulmonary and cardiovascular adverse events and air pollution, but uncertainty remains as to the extent implied by those associations although the abundance of the data. In this paper we describe an SSA (Singular Spectrum Analysis) based approach in order to decompose the time-series of particulate matter concentration into a set of exposure variables, each one representing a different timescale. We implement our methodology to investigate both acute and long-term effects of PM10PM_{10} exposure on morbidity from respiratory causes within the urban area of Bari, Italy.Comment: Published in at http://dx.doi.org/10.1214/07-EJS123 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Linear, Deterministic, and Order-Invariant Initialization Methods for the K-Means Clustering Algorithm

    Full text link
    Over the past five decades, k-means has become the clustering algorithm of choice in many application domains primarily due to its simplicity, time/space efficiency, and invariance to the ordering of the data points. Unfortunately, the algorithm's sensitivity to the initial selection of the cluster centers remains to be its most serious drawback. Numerous initialization methods have been proposed to address this drawback. Many of these methods, however, have time complexity superlinear in the number of data points, which makes them impractical for large data sets. On the other hand, linear methods are often random and/or sensitive to the ordering of the data points. These methods are generally unreliable in that the quality of their results is unpredictable. Therefore, it is common practice to perform multiple runs of such methods and take the output of the run that produces the best results. Such a practice, however, greatly increases the computational requirements of the otherwise highly efficient k-means algorithm. In this chapter, we investigate the empirical performance of six linear, deterministic (non-random), and order-invariant k-means initialization methods on a large and diverse collection of data sets from the UCI Machine Learning Repository. The results demonstrate that two relatively unknown hierarchical initialization methods due to Su and Dy outperform the remaining four methods with respect to two objective effectiveness criteria. In addition, a recent method due to Erisoglu et al. performs surprisingly poorly.Comment: 21 pages, 2 figures, 5 tables, Partitional Clustering Algorithms (Springer, 2014). arXiv admin note: substantial text overlap with arXiv:1304.7465, arXiv:1209.196

    Sparse recovery by reduced variance stochastic approximation

    Full text link
    In this paper, we discuss application of iterative Stochastic Optimization routines to the problem of sparse signal recovery from noisy observation. Using Stochastic Mirror Descent algorithm as a building block, we develop a multistage procedure for recovery of sparse solutions to Stochastic Optimization problem under assumption of smoothness and quadratic minoration on the expected objective. An interesting feature of the proposed algorithm is its linear convergence of the approximate solution during the preliminary phase of the routine when the component of stochastic error in the gradient observation which is due to bad initial approximation of the optimal solution is larger than the "ideal" asymptotic error component owing to observation noise "at the optimal solution." We also show how one can straightforwardly enhance reliability of the corresponding solution by using Median-of-Means like techniques. We illustrate the performance of the proposed algorithms in application to classical problems of recovery of sparse and low rank signals in linear regression framework. We show, under rather weak assumption on the regressor and noise distributions, how they lead to parameter estimates which obey (up to factors which are logarithmic in problem dimension and confidence level) the best known to us accuracy bounds

    Applied Harmonic Analysis and Sparse Approximation

    Get PDF
    Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations
    • …
    corecore