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Nonparametric reconstruction of a multifractal function from noisy data

Abstract : We estimate a real-valued function f of d variables, subject to additive Gaussian perturbation at noise level epsilon > 0, under L(pi)-loss, for pi >= 1. The main novelty is that f can have an extremely varying local smoothness, exhibiting a so-called multifractal behaviour. The results of Jaffard on the Frisch-Parisi conjecture suggest to link the singularity spectrum of f to Besov properties of the signal that can be handled by wavelet thresholding for denoising purposes. We prove that the optimal (minimax) rate of estimation of multifractal functions with singularity spectrum d(H) has explicit representation epsilon(2v(d(center dot),pi)), with v(d(center dot),pi) = min(H) H + (d - d(H))/pi/2H + d The minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases. In particular, the usual rate epsilon(2s/(2s+d)) is retrieved for monofractal functions (with spectrum reduced to a single value s) irrespectively of pi. More interestingly, the sparse case of estimation over single Besov balls has a new interpretation in terms of multifractal analysis.
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Submitted on : Tuesday, May 1, 2012 - 7:40:19 PM
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Arnaud Gloter, Marc Hoffmann. Nonparametric reconstruction of a multifractal function from noisy data. Probability Theory and Related Fields, Springer Verlag, 2010, 146 (1-2), pp.155--187. ⟨10.1007/s00440-008-0187-1⟩. ⟨hal-00693036⟩

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