Papers by Dominique Picard
Electronic Journal of Statistics, 2012
We consider linear inverse problems in a nonparametric statistical framework. Both the signal and... more We consider linear inverse problems in a nonparametric statistical framework. Both the signal and the operator are unknown and subject to error measurements. We establish minimax rates of convergence under squared error loss when the operator admits a blockwise singular value decomposition (blockwise SVD) and the smoothness of the signal is measured in a Sobolev sense. We construct a nonlinear procedure adapting simultaneously to the unknown smoothness of both the signal and the operator and achieving the optimal rate of convergence to within logarithmic terms. When the noise level in the operator is dominant, by taking full advantage of the blockwise SVD property, we demonstrate that the block SVD procedure overperforms classical methods based on Galerkin projection or nonlinear wavelet thresholding . We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere.
Annals of Statistics, Jun 1, 2002
Annals of Statistics, Feb 1, 2000
We present a procedure associated with nonlinear wavelet methods that provides adaptive confidenc... more We present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around f x 0 , in either a white noise model or a regression setting. A suitable modification in the truncation rule for wavelets allows construction of confidence intervals that achieve optimal coverage accuracy up to a logarithmic factor. The procedure does not require knowledge of the regularity of the unknown function f; it is also efficient for functions with a low degree of regularity.
Convergence of the Bayes posterior measure is considered in canonical statistical settings where ... more Convergence of the Bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions. A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.
We provide a new algorithm for the treatment of deconvolution on the sphere which combines the tr... more We provide a new algorithm for the treatment of deconvolution on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behaviour of our procedure for any Lp loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We also perform a numerical study which proves that the procedure shows very promising properties in practice as well. 1. Introduction. The
Let X 1 , ..., X n be a random sample from some unknown probability density f defined on a compac... more Let X 1 , ..., X n be a random sample from some unknown probability density f defined on a compact homogeneous manifold M of dimension d ≥ 1. Consider a 'needlet frame' {φ jη } describing a localised projection onto the space of eigenfunctions of the Laplace operator on M with corresponding eigenvalues less than 2 2j , as constructed in Geller and Pesenson [2010]. We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator f n (j) obtained from an empirical estimate of the needlet projection η φ jη f φ jη of f . We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density f . The confidence bands are adaptive over classes of differentiable and Hölder-continuous functions on M that attain their Hölder exponents. MSC 2000: 62G07, 60E15, 42C40 We summarize here some facts on compact homogeneous manifolds and Lie groups (see , , Faraut [2008] for general references), and the construction and essential properties of the associated needlet frame due to Geller and Mayeli [2009], , generalising the spherical case considered in Narcowich et al. [2006a]. ) the identity e ∈ G satisfies e.x = x and if d) for every g ∈ G, g = e, there exists a point x ∈ M such that g.x = x. A group G acts transitively on M if in addition for every x, y ∈ M there exists g ∈ G s.t. g.x = y.
The Annals of Statistics, 2002
Adaptive sampling schemes with multiple sampling rates have the potential to significantly improv... more Adaptive sampling schemes with multiple sampling rates have the potential to significantly improve the efficiency and effectiveness of methods for signal analysis. For example, in the case of equipment which transmits data continuously, multi-rate methods can reduce the cost of transmission. For equipment which transmits data only periodically they can reduce the costs of both storage and transmission. When multiple sampling rates are used in connection with wavelet estimators, the most natural algorithms for rate-switching are arguably those based on threshold-crossings by wavelet coefficients. In this paper we study the performance of such algorithms, and show that even simple threshold-crossing rules can achieve near-optimal convergence rates. A new mathematical model is suggested for assessing performance, combining the simplicity and familiarity of global approaches with an account of the local variation to which multi-rate sampling responds.
Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006
We consider the problem of recovering a function f , when we receive a blurred (by a linear opera... more We consider the problem of recovering a function f , when we receive a blurred (by a linear operator) and noisy version : Yε = Kf + ε Ẇ . We will have as guides 2 famous examples of such inverse problems : the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It can't be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising we have to 'inverse the operator' K, which operation not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem, there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable : this is the Singular Value Decomposition basis. On the other hand, the SVD basis might be difficult to determine and manipulate numerically, it also might not be appropriate for the accurate description of the solution with a small number of parameters. Also in many practical situations, the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures trying to keep the advantages of localisation without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD) using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithm factors for any Lp loss function, and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD bases lies in the range of Jacobi polynomials), we prove that quite a similar algorithm (NEED-VD) can be performed provided replacing the standard wavelet system by a second generation wavelet-type basis : the Needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I). It has its own interest and will act as a guide for understanding the 'true' inverse models for a reader which is unfamiliar with nonparametric statistical estimation. It can be read first. Section 1 introduces the general inverse problem and describes the examples of deconvolution and Wicksell problem. A review of standard methods is given with a special focus on SVD methods. Section 2 describes the WAVE-VD procedure. Section 3 and 4 give a description of the needlets constructions and the performances of the NEED-VD procedure.
The Annals of Statistics, 1996
Density estimation is a commonly used test case for nonparametric estimation methods. We explore ... more Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes B σpq and for a range of global L p error measures, 1 ≤ p < ∞. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p > p, some form of nonlinearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of n. A second approach, using an approximation of a Gaussian white-noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error (p = 2).
Lecture Notes in Statistics, 1998
The mathematical theory of ondelettes (wavelets) was developed by Yves Meyer and many collaborato... more The mathematical theory of ondelettes (wavelets) was developed by Yves Meyer and many collaborators about 10 years ago. It was designed for approximation of possibly irregular functions and surfaces and was successfully applied in data compression, turbulence analysis, image and signal processing. Five years ago wavelet theory progressively appeared to be a powerful framework for nonparametric statistical problems. Efficient computational implementations are beginning to surface in this second lustrum of the nineties. This book brings together these three main streams of wavelet theory. It presents the theory, discusses approximations and gives a variety of statistical applications. It is the aim of this text to introduce the novice in this field into the various aspects of wavelets. Wavelets require a highly interactive computing interface. We present therefore all applications with software code from an interactive statistical computing environment. Readers interested in theory and construction of wavelets will find here in a condensed form results that are somewhat scattered around in the research literature. A practioner will be able to use wavelets via the available software code. We hope therefore to address both theory and practice with this book and thus help to construct bridges between the different groups of scientists. This text grew out of a French-German cooperation (Séminaire Paris-Berlin, Seminar Berlin-Paris). This seminar brings together theoretical and applied statisticians from Berlin and Paris. This work originates in the first of these seminars organized in Garchy, Burgundy in 1994. We are confident that there will be future research work originating from this yearly seminar. This text would not have been possible without discussion and encouragement from colleagues in France and Germany. We would like to thank in particular
We consider the nonparametric estimation of a function that is observed in white noise after conv... more We consider the nonparametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval ( a,a). In a recent paper Johnstone et al. (2004) have developped a wavelet deconvolution algorithm (called WaveD) that can be used for "certain" boxcar kernels. For example, WaveD can be tuned to achieve near optimal rates over Besov spaces when a is a Badly Approximable (BA) irrational number. While the set of all BA's contains quadratic irrationals e.g. a = p 5 it has Lebesgue measure zero, however. In this paper we derive two tuning scenarios of WaveD that are valid for "almost all" boxcar convolution (i.e. when a 2 A where A is a full Lebesgue measure set). We propose (i) a tuning inspired from Minimax theory over Besov spaces; (ii) a tuning inspired from Maxiset theory providing similar rates as for BA numbers. Asymptotic theory informs that (i) in the worst case scenario, departure from the BA assumption,...
Probability Theory and Related Fields, 2013
Convergence of the Bayes posterior measure is considered in canonical statistical settings where ... more Convergence of the Bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions. A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.
Probability Theory and Related Fields, 2001
In the framework of denoising a function depending of a multidimensional variable (for instance a... more In the framework of denoising a function depending of a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our methodisageneralizationoftheLepski'smethodofadaptation,androughlyconsistsinchoosing the "coarsest" bandwidth such that the estimated bias is negligible. However, this notion becomes more delicate in a multidimensional setting. We will particularly focus on functions with inhomogeneous smoothness properties and especially providing a possible disparity of the inhomogeneous aspect in the different directions. We show, in particular that our method is able to exactly attain the minimax rate or to adapt to unknown degree of anisotropic smoothness up to a logarithmic factor, for a large scale of anisotropic Besov spaces.
Probability Theory and Related Fields, 2011
Let X 1 , . . . , X n be a random sample from some unknown probability density f defined on a com... more Let X 1 , . . . , X n be a random sample from some unknown probability density f defined on a compact homogeneous manifold M of dimension d ≥ 1. Consider a 'needlet frame' {φ jη } describing a localised projection onto the space of eigenfunctions of the Laplace operator on M with corresponding eigenvalues less than 2 2 j , as constructed in Geller and Pesenson (J Geom Anal 2011). We prove nonasymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator f n ( j) obtained from an empirical estimate of the needlet projection η φ jη f φ jη of f . We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density f . The confidence bands are adaptive over classes of differentiable and Hölder-continuous functions on M that attain their Hölder exponents.
Journal of the Royal Statistical Society Series B: Statistical Methodology, 2004
SummaryDeconvolution problems are naturally represented in the Fourier domain, whereas thresholdi... more SummaryDeconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with O{n log (n)2} steps. In the periodic setting, the method applies to most deconvolution problems, including certain ‘boxcar’ kernels, which are important as a model of motion blur, but having poor Fourier characteristics. Asymptotic theory informs the choice of tuning parameters and yields adaptivity properties for the method over a wide class of measures of error and classes of function. The method is tested on simulated light detection and ranging data suggested by underwater remote sensing. Both visual and numerical results show an improvement over competing approaches. Finally, the theory behind our estimation paradigm gives a complete characterization of the ‘maxiset’ of the met...
Journal of Functional Analysis, 2009
A pair of dual frames with almost exponentially localized elements (needlets) are constructed on ... more A pair of dual frames with almost exponentially localized elements (needlets) are constructed on R d + based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.
Foundations of Computational Mathematics, 2005
Let ρ be an unknown Borel measure defined on the space Z := X × Y with X ⊂ IR d and Y = [-M, M ].... more Let ρ be an unknown Borel measure defined on the space Z := X × Y with X ⊂ IR d and Y = [-M, M ]. Given a set z of m samples z i = (x i , y i ) drawn according to ρ, the problem of estimating a regression function f ρ using these samples is considered. The main focus is to understand what is the rate of approximation, measured either in expectation or probability, that can be obtained under a given prior f ρ ∈ Θ, i.e. under the assumption that f ρ is in the set Θ, and what are possible algorithms for obtaining optimal or semi-optimal (up to logarithms) results. The optimal rate of decay in terms of m is established for many priors given either in terms of smoothness of f ρ or its rate of approximation measured in one of several ways. This optimal rate is determined by two types of results. Upper bounds are established using various tools in approximation such as entropy, widths, and linear and nonlinear approximation. Lower bounds are proved using Kullback-Leibler information together with Fano inequalities and a certain type of entropy. A distinction is drawn between algorithms which employ knowledge of the prior in the construction of the estimator and those that do not. Algorithms of the second type which are universally optimal for a certain range of priors are given.
Electronic Journal of Statistics, 2013
In the present paper we consider the problem of estimating a periodic (r +1)-dimensional function... more In the present paper we consider the problem of estimating a periodic (r +1)-dimensional function f based on observations from its noisy convolution. We construct a wavelet estimator of f , derive minimax lower bounds for the L 2 -risk when f belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the "curse of dimensionality" and, hence, are higher than usual convergence rates when r is large. The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function f is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of M solutions of separate convolution equations. A limited
Electronic Journal of Statistics, 2007
We provide a new algorithm for the treatment of inverse problems which combines the traditional S... more We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed "needlets") built upon the SVD bases. We consider two different situations: the "wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the "Jacobitype" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the L 2 case, we obtain interesting rates of convergence for other Lp norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-D estimator outperforms other standard algorithms in almost all situations.
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Papers by Dominique Picard