The Experts below are selected from a list of 174 Experts worldwide ranked by ideXlab platform

Christophe Chesneau - One of the best experts on this subject based on the ideXlab platform.

  • On a Plug-In Wavelet Estimator for Convolutions of Densities
    Journal of Statistical Theory and Practice, 2014
    Co-Authors: Christophe Chesneau, Fabien Navarro
    Abstract:

    The nonparametric estimation of the m-fold convolution power of an unknown function f is considered. We introduce an Estimator based on a plug-in approach and a wavelet hard Thresholding Estimator. We explore its theoretical asymptotic performances via the mean integrated squared error assuming that f has a certain degree of smoothness. Applications and numerical examples are given for the standard density estimation problem and the deconvolution density estimation problem.

  • A general result on the performance of the wavelet hard Thresholding Estimator under $\alpha$-mixing dependence
    2014
    Co-Authors: Christophe Chesneau
    Abstract:

    In this note, we consider the estimation of an unknown function $f$ for weakly dependent data ($\alpha$-mixing) in a general setting. Our contribution is theoretical: we prove that a wavelet hard Thresholding Estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design.

  • Adaptive wavelet deconvolution for strongly mixing sequences
    2011
    Co-Authors: Christophe Chesneau
    Abstract:

    This paper studies the estimation of a density in the convolution density model from weakly dependent observations. The ordinary smooth case is considered. Adopting the minimax approach under the mean integrated square error over Besov balls, we explore the performances of two wavelet Estimators: a standard linear one based on projections and a new non-linear one based on a hard Thresholding rule. In particular, under strong mixing conditions, we prove that our hard Thresholding Estimator attains a particular rate of convergence: the optimal one in the {\it i.i.d.} case up to a logarithmic term.

  • WAVELET BLOCK Thresholding FOR DENSITY ESTIMATION IN THE PRESENCE OF BIAS
    2007
    Co-Authors: Christophe Chesneau
    Abstract:

    We consider the density estimation problem from i.i.d. biased observations. We investigate the performances of an adaptive wavelet block Thresholding Estimator via the minimax approach under the Lp risk with p>=1 over Besov balls. We prove that it achieves near optimal rates of convergence.

  • Wavelet block Thresholding for samples with random design: a minimax approach under the $L^p$ risk
    Electronic Journal of Statistics, 2007
    Co-Authors: Christophe Chesneau
    Abstract:

    We consider the regression model with (known) random design. We investigate the minimax performances of an adaptive wavelet block Thresholding Estimator under the $\mathbb{L}^p$ risk with $p\ge 2$ over Besov balls. We prove that it is near optimal and that it achieves better rates of convergence than the conventional term-by-term Estimators (hard, soft,...).

Sylvain Sardy - One of the best experts on this subject based on the ideXlab platform.

  • smooth blockwise iterative Thresholding a smooth fixed point Estimator based on the likelihood s block gradient
    Journal of the American Statistical Association, 2012
    Co-Authors: Sylvain Sardy
    Abstract:

    The proposed smooth blockwise iterative Thresholding Estimator (SBITE) is a model selection technique defined as a fixed point reached by iterating a likelihood gradient-based Thresholding function. The smooth James–Stein Thresholding function has two regularization parameters λ and ν, and a smoothness parameter s. It enjoys smoothness like ridge regression and selects variables like lasso. Focusing on Gaussian regression, we show that SBITE is uniquely defined, and that its Stein unbiased risk estimate is a smooth function of λ and ν, for better selection of the two regularization parameters. We perform a Monte Carlo simulation to investigate the predictive and oracle properties of this smooth version of adaptive lasso. The motivation is a gravitational wave burst detection problem from several concomitant time series. A nonparametric wavelet-based Estimator is developed to combine information from all captors by block-Thresholding multiresolution coefficients. We study how the smoothness parameter s tem...

  • smooth blockwise iterative Thresholding a smooth fixed point Estimator based on the likelihood s block gradient
    arXiv: Methodology, 2011
    Co-Authors: Sylvain Sardy
    Abstract:

    The proposed smooth blockwise iterative Thresholding Estimator (SBITE) is a model selection technique defined as a fixed point reached by iterating a likelihood gradient-based Thresholding function. The smooth James-Stein Thresholding function has two regularization parameters $\lambda$ and $\nu$, and a smoothness parameter $s$. It enjoys smoothness like ridge regression and selects variables like lasso. Focusing on Gaussian regression, we show that SBITE is uniquely defined, and that its Stein unbiased risk estimate is a smooth function of $\lambda$ and $\nu$, for better selection of the two regularization parameters. We perform a Monte-Carlo simulation to investigate the predictive and oracle properties of this smooth version of adaptive lasso. The motivation is a gravitational wave burst detection problem from several concomitant time series. A nonparametric wavelet-based Estimator is developed to combine information from all captors by block-Thresholding multiresolution coefficients. We study how the smoothness parameter $s$ tempers the erraticity of the risk estimate, and derive a universal threshold, an information criterion and an oracle inequality in this canonical setting.

Katsuyuki Hagiwara - One of the best experts on this subject based on the ideXlab platform.

  • Adaptive scaling for soft-Thresholding Estimator
    arXiv: Methodology, 2016
    Co-Authors: Katsuyuki Hagiwara
    Abstract:

    Soft-Thresholding is a sparse modeling method that is typically applied to wavelet denoising in statistical signal processing and analysis. It has a single parameter that controls a threshold level on wavelet coefficients and, simultaneously, amount of shrinkage for coefficients of un-removed components. This parametrization is possible to cause excess shrinkage, thus, estimation bias at a sparse representation; i.e. there is a dilemma between sparsity and prediction accuracy. To relax this problem, we considered to introduce positive scaling on soft-Thresholding Estimator, by which threshold level and amount of shrinkage are independently controlled. Especially, in this paper, we proposed component-wise and data-dependent scaling in a setting of non-parametric orthogonal regression problem including discrete wavelet transform. We call our scaling method adaptive scaling. We here employed soft-Thresholding method based on LARS(least angle regression), by which the model selection problem reduces to the determination of the number of un-removed components. We derived a risk under LARS-based soft-Thresholding with the proposed adaptive scaling and established a model selection criterion as an unbiased estimate of the risk. We also analyzed some properties of the risk curve and found that the model selection criterion is possible to select a model with low risk and high sparsity compared to a naive soft-Thresholding method. This theoretical speculation was verified by a simple numerical experiment and an application to wavelet denoising.

  • On scaling of soft-Thresholding Estimator
    Neurocomputing, 2016
    Co-Authors: Katsuyuki Hagiwara
    Abstract:

    LASSO is known to have a problem of excessive shrinkage at a sparse representation. To analyze this problem in detail, in this paper, we consider a positive scaling for soft-Thresholding Estimators that are LASSO Estimators in an orthogonal regression problem. We especially consider a non-parametric orthogonal regression problem which includes wavelet denosing. We first gave a risk (generalization error) of LARS (least angle regression) based soft-Thresholding with a single scaling parameter. We then showed that an optimal scaling value that minimizes the risk under a sparseness condition is 1 + O ( log n / n ) , where n is the number of samples. The important point is that the optimal value of scaling is larger than one. This implies that expanding soft-Thresholding Estimator shows a better generalization performance compared to a naive soft-Thresholding. This also implies that a risk of LARS-based soft-Thresholding with the optimal scaling is smaller than without scaling. We then showed their difference is O ( log n / n ) . This also shows an effectiveness of the introduction of scaling. Through simple numerical experiments, we found that LARS-based soft-Thresholding with scaling can improve both of sparsity and generalization performance compared to a naive soft-Thresholding.

Ramji Venkataramanan - One of the best experts on this subject based on the ideXlab platform.

  • Empirical Bayes Estimators for high-dimensional sparse vectors
    Information and Inference: A Journal of the IMA, 2019
    Co-Authors: K. Pavan Srinath, Ramji Venkataramanan
    Abstract:

    The problem of estimating a high-dimensional sparse vector $\boldsymbol{\theta} \in \mathbb{R}^n$ from an observation in i.i.d. Gaussian noise is considered. The performance is measured using squared-error loss. An empirical Bayes shrinkage Estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-Thresholding Estimator. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both Estimators. The results show that for large $n$, both the risk estimate and the loss function concentrate on deterministic values close to the true risk. Depending on the underlying $\boldsymbol{\theta}$, either the proposed empirical Bayes (eBayes) Estimator or soft-Thresholding may have smaller loss. We consider a hybrid Estimator that attempts to pick the better of the soft-Thresholding Estimator and the eBayes Estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid Estimator concentrates on the minimum of the losses of the two competing Estimators, and ii) the risk of the hybrid Estimator is within order $\frac{1}{\sqrt{n}}$ of the minimum of the two risks. Simulation results are provided to support the theoretical results. Finally, we use the eBayes and hybrid Estimators as denoisers in the approximate message passing (AMP) algorithm for compressed sensing, and show that their performance is superior to the soft-Thresholding denoiser in a wide range of settings.

  • Empirical Bayes Estimators for Sparse Sequences
    2018 IEEE International Symposium on Information Theory (ISIT), 2018
    Co-Authors: Pavan K. Srinath, Ramji Venkataramanan
    Abstract:

    The problem of estimating a high-dimensional sparse vector θ ∈ ℝn from an observation in i.i.d. Gaussian noise is considered. An empirical Bayes shrinkage Estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-Thresholding Estimator using squared-error loss as a measure of performance. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both Estimators. Depending on the underlying θ, either the proposed empirical Bayes (eBayes) Estimator or soft-Thresholding may have smaller loss. We consider a hybrid Estimator that attempts to pick the better of the soft-Thresholding Estimator and the eBayes Estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid Estimator concentrates on the minimum of the losses of the two competing Estimators, and ii) the risk of the hybrid Estimator is within order 1/√n of the minimum of the two risks. Simulation results are provided to support the theoretical results.

  • ISIT - Empirical Bayes Estimators for Sparse Sequences
    2018 IEEE International Symposium on Information Theory (ISIT), 2018
    Co-Authors: K. Pavan Srinath, Ramji Venkataramanan
    Abstract:

    The problem of estimating a high-dimensional sparse vector $\theta\in \mathbb{R}^{n}$ from an observation in i.i.d. Gaussian noise is considered. An empirical Bayes shrinkage Estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-Thresholding Estimator using squared-error loss as a measure of performance. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both Estimators. Depending on the underlying $\theta$ , either the proposed empirical Bayes (eBayes) Estimator or soft-Thresholding may have smaller loss. We consider a hybrid Estimator that attempts to pick the better of the soft-Thresholding Estimator and the eBayes Estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid Estimator concentrates on the minimum of the losses of the two competing Estimators, and ii) the risk of the hybrid Estimator is within order $1/\sqrt{n}$ of the minimum of the two risks. Simulation results are provided to support the theoretical results.

Harrison H. Zhou - One of the best experts on this subject based on the ideXlab platform.

  • the root unroot algorithm for density estimation as implemented via wavelet block Thresholding
    Probability Theory and Related Fields, 2010
    Co-Authors: Lawrence D Brown, Ren Zhang, Linda Zhao, Tony T Cai, Harrison H. Zhou
    Abstract:

    We propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem. This regression problem is created by binning the original observations into many small size bins, and by then applying a suitable form of root transformation to the binned data counts. In principle many common nonparametric regression Estimators could then be applied to the transformed data. We propose use of a wavelet block Thresholding Estimator in this paper. Finally, the estimated regression function is un-rooted by squaring and normalizing. The density estimation procedure achieves simultaneously three objectives: computational efficiency, adaptivity, and spatial adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this procedure. Theoretically it is shown that the Estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes. The Estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. There are three key steps in the technical argument: Poissonization, quantile coupling, and oracle risk bound for block Thresholding in the non-Gaussian setting. Some of the technical results may be of independent interest.

  • Nonparametric regression in natural exponential families
    Institute of Mathematical Statistics Collections, 2010
    Co-Authors: T. Toni Cai, Harrison H. Zhou
    Abstract:

    Theory and methodology for nonparametric regression have been particularly well developed in the case of additive homoscedastic Gaussian noise. Inspired by asymptotic equivalence theory, there have been ongoing efforts in recent years to construct explicit procedures that turn other function estimation problems into a standard nonparametric regression with Gaussian noise. Then in principle any good Gaussian nonparametric regression method can be used to solve those more complicated nonparametric models. In particular, Brown, Cai and Zhou [3] considered nonparametric regression in natural exponential families with a quadratic variance function. In this paper we extend the scope of Brown, Cai and Zhou [3] to general natural exponential families by introducing a new explicit procedure that is based on the variance stabilizing transformation. The new approach significantly reduces the bias of the inverse transformation and as a consequence it enables the method to be applicable to a wider class of exponential families. Combining this procedure with a wavelet block Thresholding Estimator for Gaussian nonparametric regression, we show that the resulting Estimator enjoys a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a broad range of Besov spaces.