sharp systolic inequalities for reeb flows on the three sphere the systolic ratio of a contact form alpha on the three sphere is the quantity begin aligned rho mathrm sys alpha frac t min alpha 2 mathrm vol s 3 alpha wedge d alpha end aligned where t min alpha is the minimal period of closed reeb orbits on s 3 alpha a zoll contact form is a contact form such that all the orbits of the corresponding reeb flow are closed and have the same period our first main result is that rho mathrm sys le 1 in a neighbourhood of the space of zoll contact forms on s 3 with equality holding precisely at zoll contact forms this implies a particular case of a conjecture of viterbo a local middle dimensional non squeezing theorem and a sharp systolic inequality for finsler metrics on the two sphere which are close to zoll ones our second main result is that rho mathrm sys is unbounded from above on the space of tight contact forms on s 3

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

spatially inhomogeneous linear inverse problems with possible singularities the objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem which to the best of the author s knowledge has never been considered previously in statistical framework but is emerging due to a variety of practical applications the special feature of the problem is that the degree of ill posedness depends not only on the scale but also on location in this case the rates of convergence are determined by the interaction of four parameters the smoothness and spatial homogeneity of the unknown function f and degrees of ill posedness and spatial inhomogeneity of operator q an interesting property here is that if operator q is weakly inhomogeneous then the rates of convergence are not influenced by spatial inhomogeneity of operator q and coincide with the rates which are usual for homogeneous linear inverse problems on the other hand if operator q is moderately or strongly inhomogeneous convergence rates are significantly affected by the degree of spatial inhomogeneity estimators obtained in the paper are based either on wavelet vaguelette decomposition if the norms of all vaguelettes are finite or on a hybrid of wavelet vaguelette decomposition and galerkin method if vaguelettes in the neighborhood of the singularity point have infinite norms the hybrid estimator is a combination of a linear part in the vicinity of the singularity point and the nonlinear block thresholding wavelet estimator elsewhere to attain adaptivity an optimal resolution level for the linear singularity affected portion of the estimator is obtained using lepskii 1990 1999 method subsequently this resolution level is used as the lowest resolution level for the nonlinear wavelet estimator we show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates the theory presented in the paper is supplemented by examples of deconvolution based on irregularly spaced observations or in the presence of a heteroscedastic noise the latter problem is also examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high

recovery algorithms for vector valued data with joint sparsity constraints vector valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually additionally different components may also exhibit common sparsity patterns recently there were introduced sparsity measures that take into account such joint sparsity patterns promoting coupling of nonvanishing components these measures are typically constructed as weighted ell 1 norms of componentwise ell q norms of frame coefficients we show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded landweber algorithms next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures the weights are interpreted as indicators of the sparsity pattern and are iteratively updated after each new application of the thresholded landweber algorithm the resulting two step algorithm is interpreted as a double minimization scheme for a suitable target functional we show its ell 2 norm convergence an implementable version of the algorithm is also formulated and its norm convergence is proven numerical experiments in color image restoration are presented

estimation of nonparametric conditional moment models with possibly nonsmooth moments this paper studies nonparametric estimation of conditional moment models in which the residual functions could be nonsmooth with respect to the unknown functions of endogenous variables it is a problem of nonparametric nonlinear instrumental variables iv estimation and a difficult nonlinear ill posed inverse problem with an unknown operator we first propose a penalized sieve minimum distance smd estimator of the unknown functions that are identified via the conditional moment models we then establish its consistency and convergence rate in strong metric allowing for possibly non compact function parameter spaces possibly non compact finite or infinite dimensional sieves with flexible lower semicompact or convex penalty or finite dimensional linear sieves without penalty under relatively low level sufficient conditions and for both mildly and severely ill posed problems we show that the convergence rates for the nonlinear ill posed inverse problems coincide with the known minimax optimal rates for the nonparametric mean iv regression we illustrate the theory by two important applications root n asymptotic normality of the plug in penalized smd estimator of a weighted average derivative of a nonparametric nonlinear iv regression and the convergence rate of a nonparametric additive quantile iv regression we also present a simulation study and an empirical estimation of a system of nonparametric quantile iv engel curves

analysis of second order differential operators on heisenberg groups ii given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

analysis of second order differential operators on heisenberg groups i given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

convergence rates of general regularization methods for statistical inverse problems and applications previously the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut off and tikhonov type estimators spectral cut off estimators achieve minimax rates for a broad range of smoothness classes and operators but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes in this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods spectral methods including the aforementioned estimators as well as many iterative methods such as nu methods and the landweber iteration the latter estimators converge at the same rate as spectral cut off but require only matrix vector products our results are applied to various problems in particular we obtain precise convergence rates for satellite gradiometry l 2 boosting and errors in variable problems

is clustering advantageous in statistical ill posed linear inverse problems in many statistical linear inverse problems one needs to recover classes of similar objects from their noisy images under an operator that does not have a bounded inverse problems of this kind appear in many areas of application routinely in such problems clustering is carried out at a pre processing step and then the inverse problem is solved for each of the cluster averages separately as a result the errors of the procedures are usually examined for the estimation step only the objective of this paper is to examine both theoretically and via simulations the effect of clustering on the accuracy of the solutions of general ill posed linear inverse problems in particular we assume that one observes x m a f m delta epsilon m m 1 cdots m where functions f m can be grouped into k classes and one needs to recover a vector function mathbf f f 1 cdots f m t we construct an estimator for mathbf f as a solution of a penalized optimization problem which corresponds to the clustering before estimation setting we derive an oracle inequality for its precision and confirm that the estimator is minimax optimal or nearly minimax optimal up to a logarithmic factor of the number of observations one of the advantages of our approach is that we do not assume that the number of clusters is known in advance subsequently we compare the accuracy of the above procedure with the precision of estimation without clustering and clustering following the recovery of each of the unknown functions separately we conclude that clustering at the pre processing step is beneficial when the problem is moderately ill posed it should be applied with extreme care when the problem is severely ill posed

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

asymptotic equivalence theory for nonparametric regression with random design this paper establishes the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function the asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the whitenoise experiments which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments with asymptotic properties identical to the true observations and given asymptotic solutions from the other the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in all other asymptotically equivalent nonparametric problems 1 introduction the purpose of this paper is to establish the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function we establish this asymptotic equivalence by constructing explicit equivalence mappings between the nonparametric regression and the white noise problems as in brown and low 1996 for their asymptotic equivalence results the equivalence mapping from the nonparametric regression to the white noise provides synthetic observations of the white noise from the nonparametric regression such that the distributions of the synthetic observations are asymptotically equivalent to those of the true observations of the white noise for any asymptotic solution to a white noise problem the application of the solution to the synthetic observations provides an asymptotic solution to the corresponding nonparametric regression problem with identical asymptotic properties likewise the equivalence mapping from the white noise produces synthetic observations of the nonparametric regression problem and synthetic asymptotic solutions to white noise problems based on those of the corresponding nonparametric regression problems the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in

laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging in the present paper we consider the problem of laplace deconvolution with noisy discrete non equally spaced observations on a finite time interval which appears in many different contexts we propose a new method for laplace deconvolution which is based on expansions of the convolution kernel the unknown function and the observed signal over laguerre functions basis which acts as a surrogate eigenfunction basis using regression setting the expansion results in a small system of linear equations with the matrix of the system being triangular and toeplitz the number m of the terms in the expansion of the estimator is controlled via complexity penalty the advantage of this methodology is that it leads to very fast computations produces no boundary effects due to extension at zero and cut off at t and provides an estimator with the risk within a logarithmic factor of m of the oracle risk under no assumptions on the model and within a constant factor of the oracle risk under mild assumptions we emphasize that in the present paper we consider the true observational model with possibly non equispaced observations which are available on a finite interval of length t and account for the bias associated with this model the study is motivated by perfusion imaging using a short injection of contrast agent a procedure which is widely applied for medical assessment of blood flows within tissues such as cancerous tumors presence of a tuning parameter a allows to choose the most advantageous time units so that both the kernel and the unknown right hand side of the equation are well represented for the further deconvolution in addition absence of boundary effects allows for an accurate estimator of the tissue blood flow parameter which is of great interest to medical doctors the methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast efficient accurate usable from a practical point of view and competitive

improved error analysis for adaptive wavelet algorithms abstract to solve a general class of elliptic operator equations cohen et al 2001 2 designed two adaptive wavelet algorithms that yield an approximate solution with error o n s in this paper we modify the error analysis for the second algorithm and optimize the upper bound numerical experiments demonstrate that our approximation has some advantages compared to that of the reference

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

oscillator spacetimes are ricci solitons abstract we consider the four dimensional oscillator group equipped with a well known one parameter family of left invariant lorentzian metrics which includes the bi invariant one gadea oubina 1999 in a suitable system of global coordinates the ricci soliton equation for these metrics translates into a system of partial differential equations solving such system we prove that all these metrics are ricci solitons in particular the bi invariant metric on the oscillator group gives rise to infinitely many ricci solitons and so also to yamabe solitons

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions 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 holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves 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

on the laplace beltrami operator on the oscillator group soit g un groupe oscillateur et soit l l operateur de laplace beltrami associe on demontre que l engrendre l algebre de tous les operateurs differentiels bi invariants sur g on construit une solution fondamentale pour l

convergence and the length spectrum abstract the author defines and analyzes the 1 k length spectra l 1 k m whose union over all k n is the classical length spectrum these new length spectra are shown to converge in the sense that lim k k 1 k m i l 1 k m 0 as m i m in the gromov hausdorff sense energy methods are introduced to estimate the shortest element of l 1 k as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension a number of gap theorems are proven including one for manifolds m n with ricci n 1 and volume close to vol s n many results in this paper hold on compact length spaces in addition to riemannian manifolds

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

semi nonparametric iv estimation of shape invariant engel curves this paper studies a shape invariant engel curve system with endogenous total expenditure in which the shape invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric engel curves we focus on the identification and estimation of both the nonparametric shapes of the engel curves and the parametric specification of the demographic scaling parameters the identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions allowing for endogeneity we establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support root n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of low level sufficient conditions our empirical application using the u k family expenditure survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of engel curves

adaptive wavelet galerkin methods for linear inverse problems we introduce and analyze numerical methods for the treatment of inverse problems based on an adaptive wavelet galerkin discretization these methods combine the theoretical advantages of the wavelet vaguelette decomposition wvd in terms of optimally adapting to the unknown smoothness of the solution together with the numerical simplicity of galerkin methods in a first step we simply combine a thresholding algorithm on the data with a galerkin inversion on a fixed linear space in a second step a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed this leads to a significant reduction of the computational cost

a proximal iteration for deconvolving poisson noisy images using sparse representations we propose an image deconvolution algorithm when the data is contaminated by poisson noise the image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms our key contributions are as follows first we handle the poisson noise properly by using the anscombe variance stabilizing transform leading to a nonlinear degradation equation with additive gaussian noise second the deconvolution problem is formulated as the minimization of a convex functional with a data fidelity term reflecting the noise properties and a nonsmooth sparsity promoting penalty over the image representation coefficients e g lscr1 norm an additional term is also included in the functional to ensure positivity of the restored image third a fast iterative forward backward splitting algorithm is proposed to solve the minimization problem we derive existence and uniqueness conditions of the solution and establish convergence of the iterative algorithm finally a gcv based model selection procedure is proposed to objectively select the regularization parameter experimental results are carried out to show the striking benefits gained from taking into account the poisson statistics of the noise these results also suggest that using sparse domain regularization may be tractable in many deconvolution applications with poisson noise such as astronomy and microscopy

nonlinear estimation for linear inverse problems with error in the operator we study two nonlinear methods for statistical linear inverse problems when the operator is not known the two constructions combine galerkin regularization and wavelet thresholding their performances depend on the underlying structure of the operator quantified by an index of sparsity we prove their rate optimality and adaptivity properties over besov classes

wavelet decomposition approaches to statistical inverse problems summary a wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise primary interest is in some function f t but data are accessible only about some linear transform corrupted by noise the usual linear methods for such inverse problems do not perform satisfactorily when f t is spatially inhomogeneous one existing nonlinear alternative is the wavelet vaguelette decomposition method based on the expansion of the unknown f t in wavelet series in the vaguelette wavelet decomposition method proposed here the observed data are expanded directly in wavelet series the performances of various methods are compared through exact risk calculations in the context of the estimation of the derivative of a function observed subject to noise a result is proved demonstrating that with a suitable universal threshold somewhat larger than that used for standard denoising problems both the wavelet based approaches have an ideal spatial adaptivity property

dynamic network models and graphon estimation in the present paper we consider a dynamic stochastic network model the objective is estimation of the tensor of connection probabilities mathbf lambda when it is generated by a dynamic stochastic block model dsbm or a dynamic graphon in particular in the context of the dsbm we derive a penalized least squares estimator widehat boldsymbol lambda of mathbf lambda and show that widehat boldsymbol lambda satisfies an oracle inequality and also attains minimax lower bounds for the risk we extend those results to estimation of mathbf lambda when it is generated by a dynamic graphon function the estimators constructed in the paper are adaptive to the unknown number of blocks in the context of the dsbm or to the smoothness of the graphon function the technique relies on the vectorization of the model and leads to much simpler mathematical arguments than the ones used previously in the stationary set up in addition all results in the paper are nonasymptotic and allow a variety of extensions

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization