Gerard Kerkyacharian - Profile on Academia.edu (original) (raw)

Papers by Gerard Kerkyacharian

Journal of the Royal Statistical Society Series B: Statistical Methodology, 2004

Capturing Ridge Functions in High Dimension from Point Queries

International audienc

In this paper, we discuss a new way of evaluating the performances of a statistical estimation pr... more In this paper, we discuss a new way of evaluating the performances of a statistical estimation procedure. This point of view consists in investigating the maximal set where a given procedure has a given rate of convergence. Although the setting is not extremely different from the minimax context, it is in a sense less pessimistic and provides a functional set which is authentically connected to the procedure and the model. We also investigate more traditional concerns about procedures: oracle inequalities. This notion becomes more difficult even to be practically defined when the loss function is not the L 2-norm. We explain the difficulties arising there, and suggest a new definition, in the cases of L p-norms and point-wise estimation. The connections between maxisets and local oracle inequalities are investigated: we prove that verifying a local oracle inequality implies that the maxiset automatically contains a prescribed set linked with the oracle inequality. We have investigated the consequences of the previous statement on well known efficient adaptive methods: Wavelet thresholding and local bandwidth selection. We can prove local oracle inequalities for these methods and draw the conclusions about there associated maxisets.

Statistics & Probability Letters, 1997

We prove here that when estimating a density, using a kernel or a linear wavelet estimate, one ca... more We prove here that when estimating a density, using a kernel or a linear wavelet estimate, one can choose the smoothing parameter such that the limit when n tends to infinity of n2/3EIIF-fn I1~ may be arbitrarily small for every density having a square integrable derivative. This choice consists in starting from the usual rate n-1/3 and then operate an oversmoothing proportional to the limit of the risk we want to obtain. Looking at the limit of the risk is another way of looking at the performances of estimators: We introduce here the maximal functional space where the results still stand. We show that this space contains the Sobolev spaces for instance. We also give a comparison with the standard minimax theory.

Journal of Statistical Planning and Inference, 2007

We consider the estimation of nonparametric regression function with long memory data and investi... more We consider the estimation of nonparametric regression function with long memory data and investigate the asymptotic rates of convergence of wavelet estimators based on block thresholding. We show that the estimators achieve optimal minimax convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of long memory noise, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes.

Universal Near Minimaxity of

We discuss a method for curve estimation based on n noisy data; one translates the empirical wave... more We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount v210g(n) . aj.,fii. The method is nearly minimax for a wide variety of loss functions----e.g. pointwise error, global error measured in LP norms, pointwise and global error in estimation of derivatives-and for a wide range of smoothness classes, including standard HOlder classes, Sobolev classes, and Bounded Variation. This is a broader near-optimality than anything previously proposed in the minimax literature. The theory underlying the method exploits a correspondence between statistical ques tions and questions of optimal recovery and information-based complexity. This paper contains a detailed proof of the result announced in Donoho, Johnstone, Kerkyacharian & Picard (1995).

Bernoulli, 2020

We consider the problem of estimating the density of observations taking values in classical or n... more We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established, which are analogous to the existing results in the classical setting of real-valued variables.

Constructive Approximation

Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associa... more Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and Hölder inequality for the PSWF heat kernel follow from the ones in the Legendre case. As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in R d. Further, we develop the related to the PSWFs of order zero smooth functional calculus, which in turn is the necessary groundwork in developing the theory of Besov and Triebel-Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel-Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel-Lizorkin spaces generated by the Legendre operator.

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.

Constructive Approximation, 2019

The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the uni... more The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in R n , and in particular on the interval, generated by classical differential operators whose eigenfunctions are algebraic polynomials. To this end we develop a general method that employs the natural relation of such operators with weighted Laplace operators on suitable subsets of Riemannian manifolds and the existing general results on heat kernels. Our general scheme allows to consider heat kernels in the weighted cases on the interval, ball, and simplex with parameters in the full range.

Journal of Fourier Analysis and Applications, 2019

We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of a doubling metric me... more We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of a doubling metric measure space in the presence of a nonnegative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel-Lizorkin spaces. Spectral multipliers for these spaces are established as well.

Studia Mathematica, 2019

Two-sided Gaussian bounds are established for the weighted heat kernels on the unit ball and simp... more Two-sided Gaussian bounds are established for the weighted heat kernels on the unit ball and simplex in R d generated by classical differential operators whose eigenfunctions are algebraic polynomials.

Constructive Approximation, 2018

We are interested in the regularity of centered Gaussian processes (Z x (ω)) x∈M indexed by compa... more We are interested in the regularity of centered Gaussian processes (Z x (ω)) x∈M indexed by compact metric spaces (M, ρ). It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x, y) = E(Z x Z y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x, y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere. Heat kernel, Gaussian processes, Besov spaces. MSC 58J35 MSC 46E35MSC 42C15MSC 43A85 Clearly, K(x, y) determines the law of all finite dimensional random variables (Z x 1 ,. .. , Z xn). Conversely, if K(x, y) is a real valued, symmetric, and positive definite function on M × M , there exists a unique Hilbert space H of functions on M (the associated RKHS), for which K is a reproducing kernel, i.e. f (x) = f, K(x, •) H , ∀f ∈ H, ∀x ∈ M (see [5], [37], [15]). Further, if (u i) i∈I is an orthonormal basis for H, then the following representation in H holds: K(x, y) = i∈I u i (x)u i (y), ∀x, y ∈ M. Therefore, if (B i (ω)) i∈I is a family of independent N (0, 1) variables, then Z x (ω) := i∈I u i (x)B i (ω) is a centered Gaussian process with covariance K(x, y). Thus, this is a version of the previous process Z x (ω). 2.2 Gaussian processes with a zest of topology We now consider the following more specific setting. Let M be a compact space and let µ be a Radon measure on (M, B) with support M and B being the Borel sigma algebra on M. Assuming that (Ω, A, P) is a probability space we let Z : (M, B) ⊗ (Ω, A) → Z x (ω) ∈ R, be a measurable map such that (Z x) x∈M is a Gaussian process. In addition, we suppose that K(x, y) is a symmetric, continuous, and positive definite function on M × M. Then obviously the operator K defined by Kf (x) := M K(x, y)f (y)dµ(y), f ∈ L 2 (M, µ), is a self-adjoint compact positive operator (even trace-class) on L 2 (M, µ). Moreover, K(L 2) ⊂ C(M), the Banach space of continuous functions on M. Let ν 1 ≥ ν 2 ≥ • • • > 0 be the sequence of eigenvalues of K repeated according to their multiplicities and let (u k) k≥1 be the sequence of respective normalized eigenfunctions: M K(x, y)u k (y)dµ(y) = ν k u k (x). The functions u k are continuous real-valued functions and the sequence (u k) k≥1 is an orthonormal basis for L 2 (M, µ). By Mercer Theorem we have the following representation: K(x, y) = k ν k u k (x)u k (y), where the convergence is uniform. Let H ⊂ L 2 (Ω, P) be the closed Gaussian space spanned by finite linear combinations of (Z x) x∈M. Clearly, interpreting the following integral as Bochner integral with value in the Hilbert space H, we have B k (ω) = 1 √ ν k M Z x (ω)u k (x)dµ(x) ∈ H.

Journal of the Royal Statistical Society: Series B (Methodological), 1995

Considerable e ort has been directed recently to develop asymptotically minimax methods in proble... more Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nite-dimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly-or exactly-minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons { sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data one translates the empirical wavelet coe cients towards the origin by a n a m o u n t p 2 log(n) = p n. T h e method is di erent from methods in common use today, is computationally practical, and is spatially adaptive thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard H older classes, Sobolev classes, and Bounded Variation. This is a m uch broader near-optimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity.

Studia Mathematica, 2017

Maximal and atomic Hardy spaces H p and H p A , 0 < p ≤ 1, are considered in the setting of a dou... more Maximal and atomic Hardy spaces H p and H p A , 0 < p ≤ 1, are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. It is shown that H p = H p A with equivalent norms.

Large deviations and the Strassen theorem in H�lder norm

Stoch Proc Appl, 1992

Lecture Notes in Statistics Vol 129

ABSTRACT

Needvd: second generation wavelets for estimation in inverse problems

Festschrift for Lucien Le Cam, 1997

We discuss a method for curve estimation based on n noisy data one translates the empirical wavel... more We discuss a method for curve estimation based on n noisy data one translates the empirical wavelet coe cients towards the origin by an amount p 2 log(n) = p n. The method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard H older classes, Sobolev classes, and Bounded Variation. This is a broader near-optimality than anything previously proposed in the minimax literature. The theory underlying the method exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity. This paper contains a detailed proof of the result announced in Donoho, Johnstone, Kerkyacharian & Picard (1995).

Transactions of the American Mathematical Society, 2014

Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are d... more Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to a heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows the development of Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifolds, and other settings.