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316
Highly sparse representations from dictionaries are unique and independent of the sparseness measure
, 2003
"... ..."
Fast rates for support vector machines using gaussian kernels
 Ann. Statist
, 2004
"... We establish learning rates up to the order of n −1 for support vector machines with hinge loss (L1SVMs) and nontrivial distributions. For the stochastic analysis of these algorithms we use recently developed concepts such as Tsybakov’s noise assumption and local Rademacher averages. Furthermore we ..."
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Cited by 52 (7 self)
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We establish learning rates up to the order of n −1 for support vector machines with hinge loss (L1SVMs) and nontrivial distributions. For the stochastic analysis of these algorithms we use recently developed concepts such as Tsybakov’s noise assumption and local Rademacher averages. Furthermore we introduce a new geometric noise condition for distributions that is used to bound the approximation error of Gaussian kernels in terms of their widths. 1
A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV, L²) DECOMPOSITIONS
 MULTISCALE MODEL. SIMUL.
, 2004
"... We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v= ..."
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Cited by 49 (8 self)
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We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v=f ‖u‖X + λ0‖v ‖ p} Y. Such minimizers are standard tools for image manipulations
Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
 PROBABILITY SURVEYS
, 2005
"... This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions. That paper appeared in 1986 in a book containing survey papers on various types of dependence conditions and the limit theory under them. The survey here will include pa ..."
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Cited by 45 (0 self)
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This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions. That paper appeared in 1986 in a book containing survey papers on various types of dependence conditions and the limit theory under them. The survey here will include part (but not all) of the material in [18], and will also describe some relevant material that was not in that paper, especially some new discoveries and developments that have occurred since that paper was published. (Much of the new material described here involves “interlaced ” strong mixing conditions, in which the index sets are not restricted to “past ” and “future.”) At various places in this survey, open problems will be posed. There is a large literature on basic properties of strong mixing conditions. A survey such as this cannot do full justice to it. Here are a few references on important topics not covered in this survey. For the approximation of mixing sequences by martingale differences, see e.g. the book by Hall and Heyde [80]. For the direct approximation of mixing random variables by independent ones,
The Distribution of Rademacher Sums
, 1989
"... We find upper and lower bounds for Pr ( P x n t), where x 1 , x 2 ; : : : are real numbers. We express the answer in terms of the Kinterpolation norm from the theory of interpolation of Banach spaces. ..."
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Cited by 21 (8 self)
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We find upper and lower bounds for Pr ( P x n t), where x 1 , x 2 ; : : : are real numbers. We express the answer in terms of the Kinterpolation norm from the theory of interpolation of Banach spaces.
Nonlinear approximation with dictionaries. I. Direct estimates
 J. Fourier Anal. Appl
, 2004
"... original: 10.1007/s003650050621x publication is available at springerlink.com with DOI: 10.1007/s003650050621x ..."
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Cited by 20 (4 self)
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original: 10.1007/s003650050621x publication is available at springerlink.com with DOI: 10.1007/s003650050621x
Nonlinear piecewise polynomial approximation beyond Besov spaces
 Appl. Comput. Harmonic Anal
"... We study nonlinear nterm approximation in Lp(R2) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three famili ..."
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Cited by 19 (4 self)
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We study nonlinear nterm approximation in Lp(R2) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel nested triangulations. We call them Bspaces because they can be viewed as generalizations of Besov spaces. We use the Bspaces to prove Jackson and Bernstein estimates for nterm piecewise polynomial approximation and consequently characterize the corresponding approximation spaces by interpolation. We also develop methods for nterm piecewise polynomial approximation which capture the rates of the best approximation.
How to compare different loss functions and their risks
, 2006
"... Many learning problems are described by a risk functional which in turn is defined by a loss function, and a straightforward and widelyknown approach to learn such problems is to minimize a (modified) empirical version of this risk functional. However, in many cases this approach suffers from subst ..."
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Cited by 16 (2 self)
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Many learning problems are described by a risk functional which in turn is defined by a loss function, and a straightforward and widelyknown approach to learn such problems is to minimize a (modified) empirical version of this risk functional. However, in many cases this approach suffers from substantial problems such as computational requirements in classification or robustness concerns in regression. In order to resolve these issues many successful learning algorithms try to minimize a (modified) empirical risk of a surrogate loss function, instead. Of course, such a surrogate loss must be “reasonably related ” to the original loss function since otherwise this approach cannot work well. For classification good surrogate loss functions have been recently identified, and the relationship between the excess classification risk and the excess risk of these surrogate loss functions has been exactly described. However, beyond the classification problem little is known on good surrogate loss functions up to now. In this work we establish a general theory that provides powerful tools for comparing excess risks of different loss functions. We then apply this theory to several learning problems including (costsensitive) classification, regression, density estimation, and density level detection.
Commutator structure of operator ideals
, 1997
"... For any ideals I and J in the algebra of bounded operators, BðHÞ; on a separable infinite dimensional Hilbert space H; we determine the commutator space I; J SN r1 I; Jr where ..."
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Cited by 15 (8 self)
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For any ideals I and J in the algebra of bounded operators, BðHÞ; on a separable infinite dimensional Hilbert space H; we determine the commutator space I; J SN r1 I; Jr where