Article

# Regression Level Set Estimation Via Cost-Sensitive Classification

Dept. of Electr. Eng. & Comput. Sci., Michigan Univ., Ann Arbor, MI

IEEE Transactions on Signal Processing (Impact Factor: 2.81). 07/2007; DOI: 10.1109/TSP.2007.893758 Source: DBLP

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**ABSTRACT:**We consider the problem of estimating the region on which a non-parametric regression function is at its baseline level in two dimensions. The baseline level typically corresponds to the minimum/maximum of the function and estimating such regions or their complements is pertinent to several problems arising in edge estimation, environmental statistics, fMRI and related fields. We assume the baseline region to be convex and estimate it via fitting a `stump' function to approximate $p$-values obtained from tests for deviation of the regression function from its baseline level. The estimates, obtained using an algorithm originally developed for constructing convex contours of a density, are studied in two different sampling settings, one where several responses can be obtained at a number of different covariate-levels (dose-response) and the other involving limited number of response values per covariate (standard regression). The shape of the baseline region and the smoothness of the regression function at its boundary play a critical role in determining the rate of convergence of our estimate: for a regression function which is `p-regular' at the boundary of the convex baseline region, our estimate converges at a rate $N^{2/(4p+3)}$ in the dose-response setting, $N$ being the total budget, and its analogue in the standard regression setting converges at a rate of $N^{1/(2p+2)}$. Extensions to non-convex baseline regions are explored as well.12/2013; -
##### Conference Paper: Learning to satisfy

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**ABSTRACT:**This paper investigates a class of learning problems called learning satisfiability (LSAT) problems, where the goal is to learn a set in the input (feature) space that satisfies a number of desired output (label/response) constraints. LSAT problems naturally arise in many applications in which one is interested in the class of inputs that produce desirable outputs, rather than simply a single optimum. A distinctive aspect of LSAT problems is that the output behavior is assessed only on the solution set, whereas in most statistical learning problems output behavior is evaluated over the entire input space. We present a novel support vector machine (SVM) algorithm for solving LSAT problems and apply it to a synthetic data set to illustrate the impact of the LSAT formulation.Acoustics, Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on; 05/2008 -
##### Article: Level set estimation from projection measurements: Performance guarantees and fast computation

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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; · 2.97 Impact Factor

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