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16
Regularization on graphs with functionadapted diffusion process
, 2006
"... Harmonic analysis and diffusion on discrete data has been shown to lead to stateoftheart algorithms for machine learning tasks, especially in the context of semisupervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learn ..."
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Cited by 26 (5 self)
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Harmonic analysis and diffusion on discrete data has been shown to lead to stateoftheart algorithms for machine learning tasks, especially in the context of semisupervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learned, interpolated, etc.) are smooth with respect to the geometry of the data. In this paper we present a method for modifying the given geometry so the function(s) to be studied are smoother with respect to the modified geometry, and thus more amenable to treatment using harmonic analysis methods. Among the many possible applications, we consider the problems of image denoising and transductive classification. In both settings, our approach improves on standard diffusion based methods.
Polynomial operators and local smoothness classes on the unit interval
 Journal of Approximation Theory
"... We prove the existence of quadrature formulas exact for integrating high degree polynomials with respect to Jacobi weights based on scattered data on the unit interval. We also obtain a characterization of local Besov spaces using the coefficients of a tight frame expansion. 1 ..."
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Cited by 19 (10 self)
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We prove the existence of quadrature formulas exact for integrating high degree polynomials with respect to Jacobi weights based on scattered data on the unit interval. We also obtain a characterization of local Besov spaces using the coefficients of a tight frame expansion. 1
Polynomial operators for spectral approximation of piecewise analytic functions
, 2009
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"... quadrature formula for diffusion polynomials corresponding to a generalized heat kernel ..."
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quadrature formula for diffusion polynomials corresponding to a generalized heat kernel
Eignets for function approximation on manifolds
, 909
"... Let X be a compact, smooth, connected, Riemannian manifold without boundary, G: X × X → R be P a kernel. Analogous to a radial basis function network, an eignet is an expression of the form M j=1 ajG(◦, yj), where aj ∈ R, yj ∈ X, 1 ≤ j ≤ M. We describe a deterministic, universal algorithm for constr ..."
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Let X be a compact, smooth, connected, Riemannian manifold without boundary, G: X × X → R be P a kernel. Analogous to a radial basis function network, an eignet is an expression of the form M j=1 ajG(◦, yj), where aj ∈ R, yj ∈ X, 1 ≤ j ≤ M. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in L p (µ; X) for a general class of measures µ and kernels G. Our algorithm yields linear operators. Using the minimal separation amongst the centers yj as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every individual function. We also give estimates on the coefficients aj in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.