Regression Level Set Estimation Via Cost-Sensitive Classification

Department of Electrical and Computer Engineering , Rice University, Houston, Texas, United States
IEEE Transactions on Signal Processing (Impact Factor: 2.79). 07/2007; 55(6):2752 - 2757. DOI: 10.1109/TSP.2007.893758
Source: DBLP


Regression level set estimation is an important yet understudied learning task. It lies somewhere between regression function estimation and traditional binary classification, and in many cases is a more appropriate setting for questions posed in these more common frameworks. This note explains how estimating the level set of a regression function from training examples can be reduced to cost-sensitive classification. We discuss the theoretical and algorithmic benefits of this learning reduction, demonstrate several desirable properties of the associated risk, and report experimental results for histograms, support vector machines, and nearest neighbor rules on synthetic and real data

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    • "Let x ∈ R p represent our signal of interest. A γ-level set of x is defined as the set of locations where the value of the signal x exceeds some specified threshold γ; i.e. S * = S * (γ) = {j : x(j) ≥ γ}, j = 1, ..., p. Identification of level sets plays a crucial role in a variety of applications such as medical imaging where, for example, level sets can indicate presence of pathologically significant features such as tumors [1] [2]. "
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    ABSTRACT: Estimating the level set of a signal from measurements is a task that arises in a variety of fields, including medical imaging, astronomy, and digital elevation mapping. Motivated by scenarios where accurate and complete measurements of the signal may not available, we examine here a simple procedure for estimating the level set of a signal from highly incomplete measurements, which may additionally be corrupted by additive noise. The proposed procedure is based on box-constrained Total Variation (TV) regularization. We demonstrate the performance of our approach, relative to existing state-of-the-art techniques for level set estimation from compressive measurements, via several simulation examples.
    Proceedings / ICIP ... International Conference on Image Processing 10/2012; DOI:10.1109/ICIP.2012.6467424
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    • "In that work, the authors discuss tree-based techniques along the lines of [36] [45] and provide universal consistency results and rates of convergence. Inspired by the near optimality of the tree-based methods in performing level set estimation, we build on some of the key ideas of [36] [45] [34] to attack the level set estimation problem for the case of y = Af + n. "
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    ABSTRACT: Estimation of the level set of a function (i.e., regions where the function exceeds some value) is an important problem with applications in digital elevation mapping, medical imaging, astronomy, etc. In many applications, the function of interest is not observed directly. Rather, it is acquired through (linear) projection measurements, such as tomographic projections, interferometric measurements, coded-aperture measurements, and random projections associated with compressed sensing. This paper describes a new methodology for rapid and accurate estimation of the level set from such projection measurements. The key defining characteristic of the proposed method, called the projective level set estimator, is its ability to estimate the level set from projection measurements without an intermediate reconstruction step. This leads to significantly faster computation relative to heuristic "plug-in" methods that first estimate the function, typically with an iterative algorithm, and then threshold the result. The paper also includes a rigorous theoretical analysis of the proposed method, which utilizes the recent results from the non-asymptotic theory of random matrices results from the literature on concentration of measure and characterizes the estimator's performance in terms of geometry of the measurement operator and 1-norm of the discretized function.
    SIAM Journal on Imaging Sciences 09/2012; 6(4). DOI:10.1137/120891927 · 2.27 Impact Factor
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    • "Müller (1993) en parlebrì evement dans son survey. On peutégalement citer les travaux plus récents de Cavalier (1997), Scott et Davenport (2007), et Willett et Nowak (2007). L'estimateur proposé par Cavalier est fondé sur la maximisation de la masse en excès, et adapte celui proposé par Tsybakov (1997) dans le cas de la fonction de densité. "
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    ABSTRACT: Soit $(X,Y)$ un couple aléatoire à valeurs dans $\Lambda\times J$, où $\Lambda\subset\mathbbR^d$ et $J\subset\mathbbR$ sont supposés bornés. Nous construisons un estimateur plug-in des ensembles de niveaux de la fonction de régression $r$ de $Y$ sur $X$, à partir d'un estimateur à noyaux de $r$. Nous obtenons une vitesse de convergence du même ordre que celle obtenue par Cadre dans le cas de la densité. Nous discutons ensuite les résultats obtenus sur des données simulées.
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