476 research outputs found

    Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

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    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

    Learning Price-Elasticity of Smart Consumers in Power Distribution Systems

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    Demand Response is an emerging technology which will transform the power grid of tomorrow. It is revolutionary, not only because it will enable peak load shaving and will add resources to manage large distribution systems, but mainly because it will tap into an almost unexplored and extremely powerful pool of resources comprised of many small individual consumers on distribution grids. However, to utilize these resources effectively, the methods used to engage these resources must yield accurate and reliable control. A diversity of methods have been proposed to engage these new resources. As opposed to direct load control, many methods rely on consumers and/or loads responding to exogenous signals, typically in the form of energy pricing, originating from the utility or system operator. Here, we propose an open loop communication-lite method for estimating the price elasticity of many customers comprising a distribution system. We utilize a sparse linear regression method that relies on operator-controlled, inhomogeneous minor price variations, which will be fair to all the consumers. Our numerical experiments show that reliable estimation of individual and thus aggregated instantaneous elasticities is possible. We describe the limits of the reliable reconstruction as functions of the three key parameters of the system: (i) ratio of the number of communication slots (time units) per number of engaged consumers; (ii) level of sparsity (in consumer response); and (iii) signal-to-noise ratio.Comment: 6 pages, 5 figures, IEEE SmartGridComm 201

    Analysis of A Nonsmooth Optimization Approach to Robust Estimation

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    In this paper, we consider the problem of identifying a linear map from measurements which are subject to intermittent and arbitarily large errors. This is a fundamental problem in many estimation-related applications such as fault detection, state estimation in lossy networks, hybrid system identification, robust estimation, etc. The problem is hard because it exhibits some intrinsic combinatorial features. Therefore, obtaining an effective solution necessitates relaxations that are both solvable at a reasonable cost and effective in the sense that they can return the true parameter vector. The current paper discusses a nonsmooth convex optimization approach and provides a new analysis of its behavior. In particular, it is shown that under appropriate conditions on the data, an exact estimate can be recovered from data corrupted by a large (even infinite) number of gross errors.Comment: 17 pages, 9 figure

    Doubly Robust Smoothing of Dynamical Processes via Outlier Sparsity Constraints

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    Coping with outliers contaminating dynamical processes is of major importance in various applications because mismatches from nominal models are not uncommon in practice. In this context, the present paper develops novel fixed-lag and fixed-interval smoothing algorithms that are robust to outliers simultaneously present in the measurements {\it and} in the state dynamics. Outliers are handled through auxiliary unknown variables that are jointly estimated along with the state based on the least-squares criterion that is regularized with the â„“1\ell_1-norm of the outliers in order to effect sparsity control. The resultant iterative estimators rely on coordinate descent and the alternating direction method of multipliers, are expressed in closed form per iteration, and are provably convergent. Additional attractive features of the novel doubly robust smoother include: i) ability to handle both types of outliers; ii) universality to unknown nominal noise and outlier distributions; iii) flexibility to encompass maximum a posteriori optimal estimators with reliable performance under nominal conditions; and iv) improved performance relative to competing alternatives at comparable complexity, as corroborated via simulated tests.Comment: Submitted to IEEE Trans. on Signal Processin
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