Abstract
We estimate a real-valued function f of d variables, subject to additive Gaussian perturbation at noise level \({\varepsilon > 0}\), under L π-loss, for π ≥ 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 \({\varepsilon^{2v(d({\bullet}),\pi)}}\) , with
The minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases. In particular, the usual rate \({\varepsilon^{2s/(2s+d)}}\) is retrieved for monofractal functions (with spectrum reduced to a single value s) irrespectively of π. More interestingly, the sparse case of estimation over single Besov balls has a new interpretation in terms of multifractal analysis.
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Gloter, A., Hoffmann, M. Nonparametric reconstruction of a multifractal function from noisy data. Probab. Theory Relat. Fields 146, 155 (2010). https://doi.org/10.1007/s00440-008-0187-1
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DOI: https://doi.org/10.1007/s00440-008-0187-1
Keywords
- Multifractal analysis
- Frisch–Parisi conjecture
- Wavelet threshold algorithm
- Minimax rate of convergence
Mathematics Subject Classification (2000)
- 62G05
- 62G20