
Optimal sampling and Christoffel functions on general domains
We consider the problem of reconstructing an unknown function u∈ L^2(D,μ...
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Sequential sampling for optimal weighted least squares approximations in hierarchical spaces
We consider the problem of approximating an unknown function u∈ L^2(D,ρ)...
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Optimal sampling strategies for multivariate function approximation on general domains
In this paper, we address the problem of approximating a multivariate fu...
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Adaptive Sampling for Convex Regression
In this paper, we introduce the first principled adaptivesampling proce...
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Unambiguous DNFs and AlonSaksSeymour
We exhibit an unambiguous kDNF formula that requires CNF width Ω̃(k^2),...
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Optimal Envelope Approximation in Fourier Basis with Applications in TV White Space
Lowpass envelope approximation of smooth continuousvariable signals are...
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Optimal Document Exchange and New Codes for Small Number of Insertions and Deletions
This paper gives a communicationoptimal document exchange protocol and ...
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Optimal pointwise sampling for L^2 approximation
Given a function u∈ L^2=L^2(D,μ), where D⊂ℝ^d and μ is a measure on D, and a linear subspace V_n⊂ L^2 of dimension n, we show that nearbest approximation of u in V_n can be computed from a nearoptimal budget of Cn pointwise evaluations of u, with C>1 a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted leastsquares method, and the error is assessed in expected L^2 norm. This result improves on the results in [6,8] which require a sampling budget that is suboptimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class 𝒦⊂ L^2 that the sampling number ρ_Cn^ rand(𝒦)_L^2 in the randomized setting is dominated by the Kolmogorov nwidth d_n(𝒦)_L^2. While our result shows the existence of a randomized sampling with such nearoptimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm.
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