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16
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
, 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
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Cited by 1513 (20 self)
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Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy ɛ? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal f ∈ F decay like a powerlaw (or if the coefficient sequence of f in a fixed basis decays like a powerlaw), then it is possible to reconstruct f to within very high accuracy from a small number of random measurements. typical result is as follows: we rearrange the entries of f (or its coefficients in a fixed basis) in decreasing order of magnitude f  (1) ≥ f  (2) ≥... ≥ f  (N), and define the weakℓp ball as the class F of those elements whose entries obey the power decay law f  (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are Ndimensional Gaussian
Frontiers of stochastically nondominated portfolios
 Econometrica
, 2003
"... Abstract. We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose mean–risk models which are solvable by linear programming and generate portfolios whose returns are nondominated in the sense of secondorder ..."
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Cited by 21 (3 self)
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Abstract. We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose mean–risk models which are solvable by linear programming and generate portfolios whose returns are nondominated in the sense of secondorder stochastic dominance. Next, we develop a specialized parametric method for recovering the entire mean–risk efficient frontiers of these models and we illustrate its operation on a large data set involving thousands of assets and realizations. 1.
An Overview of Median and Stack Filtering
 Circuits, Systems, and Signal Processing, Special issue on Median and Morphological Filtering
, 1992
"... Abstract. Within the last two decades a small group of researchers has built a useful, nontrivial theory of nonlinear signal processing around the medianrelated filters known as rankorder filters, orderstatistic filters, weighted median filters, and stack filters. This required significant effort ..."
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Cited by 11 (2 self)
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Abstract. Within the last two decades a small group of researchers has built a useful, nontrivial theory of nonlinear signal processing around the medianrelated filters known as rankorder filters, orderstatistic filters, weighted median filters, and stack filters. This required significant effort to overcome the bias, both in education and research, toward linear theory, which has been dominant since the days of Fourier, Laplace, and "Convolute." We trace the development of this theory of nonlinear filtering from its beginnings in the study of noiseremoval properties and structural behavior of the median filter to the recently developed theory of optimal stack filtering. The theory of stack filtering provides a point of view which unifies many different filter classes, including morphological filters, so it is discussed in detail. Of particular importance is the way this theory has brought together, in a single analytical framework, both the estimationbased and the structuralbased approaches to the design of these filters. Some recent applications of median and stack filters are provided to demonstrate the effectiveness of this approach to nonlinear filtering. They include: the design of an optimal stack filter for image restoration; the use of vector median filters to attenuate impulsive noise in color images and to eliminate cross luminance and cross color in TV images; and the use of medianbased filters for image sequence coding, reconstruction, and scan rate conversion in normal TV and HDTV systems. 1.
MESTIMATION OF LINEAR MODELS WITH DEPENDENT ERRORS
, 2008
"... Abstract: We study the asymptotic behavior of Mestimates of regression parameters in multiple linear models where errors are dependent random variables. A Bahadur representation of the Mestimates is derived and a central limit theorem is established. The results are applied to linear models with e ..."
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Cited by 8 (1 self)
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Abstract: We study the asymptotic behavior of Mestimates of regression parameters in multiple linear models where errors are dependent random variables. A Bahadur representation of the Mestimates is derived and a central limit theorem is established. The results are applied to linear models with errors being shortrange dependent linear processes, heavytailed linear processes and some widely used nonlinear time series. 1
Understanding Stock Market Volatility A Business Cycle Perspective ∗
, 2008
"... One of the most prominent features of the U.S. stock market is the close connection between aggregate stock market volatility and the development of the business cycle. Figure 1 depicts the statistical relation between stock market volatility and the industrial production growth rate over the last s ..."
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Cited by 2 (0 self)
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One of the most prominent features of the U.S. stock market is the close connection between aggregate stock market volatility and the development of the business cycle. Figure 1 depicts the statistical relation between stock market volatility and the industrial production growth rate over the last sixty years, which shows that stock volatility is largely countercyclical, being larger in bad times than
Nearly optimal signal recovery from random projections: Universal encoding strategies?
 IEEE TRANS. INFO. THEORY
, 2006
"... Suppose we are given a vector f in a class F, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision in the Euclidean (`2) metric? This paper shows that if the objects of interest are sparse in a fixed ..."
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Cited by 1 (0 self)
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Suppose we are given a vector f in a class F, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision in the Euclidean (`2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector jfj (or of its coefficients in a fixed basis) obeys jfj(n) R 1 n01=p, where R>0 and p>0. Suppose that we take measurements yk = hf; Xki;k =1;...;K, where the Xk are Ndimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0 < p < 1 and with overwhelming probability, our reconstruction f] , defined as the solution to the constraints
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"... When LP is not a good idea – using structure in polyhedral optimization problems M.R.Osborne Mathematical Sciences Institute, Australian National University, ACT 0200, Australia Abstract. It has been known for almost 50 years [15] that the discrete l1 approximation problem can be solved by linear pr ..."
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When LP is not a good idea – using structure in polyhedral optimization problems M.R.Osborne Mathematical Sciences Institute, Australian National University, ACT 0200, Australia Abstract. It has been known for almost 50 years [15] that the discrete l1 approximation problem can be solved by linear programming. However, improved algorithms involve a step which can be interpreted as a line search, and which is not part of the standard solution procedures. This is the simplest example of a class of problems with a structure distinctly more complicated than that of the socalled nondegenerate linear programs. Our aim is to uncover this structure for these more general polyhedral functions and to show that it can be used it to develop what are recognizably algorithms of simplicial type for minimizing them. A key component of this work is a compact description of polyhedral convex functions described in some detail in [11], and this can be applied also in the development of active set type methods in polyhedral function constrained optimization problems. Applications include the development of new algorithms for problems which include problems in statistical estimation and data mining. 1