Hyperbolic Wavelet Thresholding Methods and the Curse of Dimensionality through the Maxiset Approach

Autin, Florent; Claeskens, Gerda; Freyermuth, Jean-Marc

doi:10.2139/ssrn.2165693

Cited by 7 publications

(12 citation statements)

References 32 publications

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“…The reader is also referred to the recent work of F. Autin, G. Claeskens, J.M. Freyermuth [3] where this problem is considered from the maxiset point of view. Other interesting applications of hyperbolic analysis can also be founded in [5], [4], [8], [6] and [7] where hyperbolic wavelet decompositions of Fractional Brownian Sheets and Linear Fractional Stable Sheets are given and are used to prove many sample paths properties of these random fields (smoothness properties, Hausdorff dimension of the graph).…”

Section: Introductionmentioning

confidence: 99%

The hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic fields

Abry¹,

Clausel²,

Jaffard³

et al. 2015

Rev. Mat. Iberoam.

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Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.

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Section: Introductionmentioning

confidence: 99%

The hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic fields

Abry¹,

Clausel²,

Jaffard³

et al. 2015

Rev. Mat. Iberoam.

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“…the anisotropic Besov classes B α s,t (M ) where parameters α and s are now both q-dimensional vectors. Such classes were rigorously defined in [Besov et al, 1979] and the multivariate wavelet thresholding procedure for a multivariate anisotropic Gaussian white noise model was proposed in [Neumann, 2000]; for a more recent approach to this topic see also [Autin et al, 2014]. The next sensible step is to allow for a function f to belong to an anisotropic functional class and to obtain an estimator of f that is, again, adaptive to a range of such functional classes and robust with respect to a range of distributions of the design vector X i and of the distributions of random errors ξ i .…”

Section: Discussion and A Future Researchmentioning

confidence: 99%

Robust functional estimation in the multivariate partial linear model

Levine

2018

Ann Inst Stat Math

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We consider the problem of adaptive estimation of the functional component in a multivariate partial linear model where the argument of the function is defined on a q-dimensional grid. Obtaining an adaptive estimator of this functional component is an important practical problem in econometrics where exact distributions of random errors and the parametric component are mostly unknown and cannot safely assumed to be normal. An estimator of the functional component that is adaptive in the mean squared sense over the wide range of multivariate Besov classes and robust to a wide choice of distributions of the linear component and random errors is constructed. It is also shown that the same estimator is locally adaptive over the same range of Besov classes and robust over large collections of distributions of the linear component and random errors as well. At any fixed point, this estimator also attains a local adaptive minimax rate. The procedure needed to obtain such an estimator turns out to depend on the choice of the right shrinkage approach in the wavelet domain. We show that one possible approach is to use the multivariate version of the classical BlockJS method. The multivariate version of BlockJS is developed in the manuscript and is shown to represent an independent interest. Finally, the Besov space scale over which the proposed estimator is locally adaptive is shown to depend on the dimensionality of the domain of the functional component; the higher the dimension, the larger the smoothness indicator of Besov spaces must be.

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“…The following theorem is a particular case of the one given in Autin et al (2014a). Since the authors have omitted the proof in the general case, we propose the one in our specific case in Appendix.…”

Section: Maxiset Of the Hyperbolic Hard Thresholding Estimatormentioning

confidence: 99%

“…In the univariate setting, it has been proven pertinent from both a minimax (Cai, 1999(Cai, , 2002 and a maxiset (Autin, 2008a) point of view to choose a block of neighboring coefficients of a size proportional to log ε −1 . Autin et al (2014a) give a precise specification for the length of the blocks in order to avoid situations where the number of blocks at a scale j may not divide 2 j in an integer number. We extend that idea to the context of hyperbolic wavelet estimators that requires to calibrate the length of the block l i ε within each orientation {i, i = 0} w.r.t the primary resolution scales.…”

Section: Maxiset Of Hyperbolic Block Thresholding Estimatormentioning

confidence: 99%

Asymptotic Performance of Projection Estimators in Standard and Hyperbolic Wavelet Bases

2015

Self Cite

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We provide a novel treatment of the ability of the standard (wavelet-tensor) and of the hyperbolic (tensor product) wavelet bases to build nonparametric estimators of multivariate functions. First, we give new results about the limitations of wavelet estimators based on the standard wavelet basis regarding their inability to optimally reconstruct functions with anisotropic smoothness. Next, we provide optimal or near optimal rates at which both linear and non-linear hyperbolic wavelet estimators are well-suited to reconstruct functions from anisotropic Besov spaces and subsequently we characterize the set of all the functions that are well reconstructed by these methods with respect to these rates. As a first main result, we furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, in particular by showing a stronger exposure of linear methods to the curse of dimensionality. Second, we propose an adaptation of the well known block thresholding method to a hyperbolic wavelet basis and show its ability to estimate functions with anisotropic smoothness at the optimal minimax rate. Therefore, we prove the pertinence of horizontal information pooling even in high dimensional settings. Numerical experiments illustrate the finite samples properties of the studied estimators.

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Hyperbolic Wavelet Thresholding Methods and the Curse of Dimensionality through the Maxiset Approach

Cited by 7 publications

References 32 publications

The hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic fields

The hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic fields

Robust functional estimation in the multivariate partial linear model

Asymptotic Performance of Projection Estimators in Standard and Hyperbolic Wavelet Bases

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