Solution of linear ill-posed problems using overcomplete dictionaries

In the present paper we consider application of overcomplete dictionaries to solution of general ill-posed linear inverse problems. Construction of an adaptive optimal solution for such problems usually relies either on a singular value decomposition or representation of the solution via an orthonormal basis. The shortcoming of both approaches lies in the fact that, in many situations, neither the eigenbasis of the linear operator nor a standard orthonormal basis constitutes an appropriate collection of functions for sparse representation of the unknown function. In the context of regression problems, there have been an enormous amount of effort to recover an unknown function using an overcomplete dictionary. One of the most popular methods, Lasso, is based on minimizing the empirical likelihood and requires stringent assumptions on the dictionary, the, so called, compatibility conditions. While these conditions may be satisfied for the original dictionary functions, they usually do not hold for their images due to contraction imposed by the linear operator. In what follows, we bypass this difficulty by a novel approach which is based on inverting each of the dictionary functions and matching the resulting expansion to the true function, thus, avoiding unrealistic assumptions on the dictionary and using Lasso in a predictive setting. We examine both the white noise and the observational model formulations and also discuss how exact inverse images of the dictionary functions can be replaced by their approximate counterparts. Furthermore, we show how the suggested methodology can be extended to the problem of estimation of a mixing density in a continuous mixture. For all the situations listed above, we provide the oracle inequalities for the risk in a finite sample setting. Simulation studies confirm good computational properties of the Lasso-based technique

Frequentist optimality of Bayesian wavelet shrinkage rules for Gaussian and non-Gaussian noise

The present paper investigates theoretical performance of various Bayesian wavelet shrinkage rules in a nonparametric regression model with i.i.d. errors which are not necessarily normally distributed. The main purpose is comparison of various Bayesian models in terms of their frequentist asymptotic optimality in Sobolev and Besov spaces. We establish a relationship between hyperparameters, verify that the majority of Bayesian models studied so far achieve theoretical optimality, state which Bayesian models cannot achieve optimal convergence rate and explain why it happens.Comment: Published at http://dx.doi.org/10.1214/009053606000000128 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Non-asymptotic approach to varying coefficient model

In the present paper we consider the varying coefficient model which represents a useful tool for exploring dynamic patterns in many applications. Existing methods typically provide asymptotic evaluation of precision of estimation procedures under the assumption that the number of observations tends to infinity. In practical applications, however, only a finite number of measurements are available. In the present paper we focus on a non-asymptotic approach to the problem. We propose a novel estimation procedure which is based on recent developments in matrix estimation. In particular, for our estimator, we obtain upper bounds for the mean squared and the pointwise estimation errors. The obtained oracle inequalities are non-asymptotic and hold for finite sample size

On convergence rates equivalency and sampling strategies in functional deconvolution models

Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its real-life discrete counterpart over a wide range of Besov balls and for the $L^2$-risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these models irregular. We formulate the conditions when each of these situations takes place. In the regular case, we not only point out the number and the selection of sampling points which deliver the fastest convergence rates in the discrete model but also investigate when, in the case of an arbitrary sampling scheme, the convergence rates in the continuous model coincide or do not coincide with the convergence rates in the discrete model. We also study what happens if one chooses a uniform, or a more general pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement of the continuous model.Comment: Published in at http://dx.doi.org/10.1214/09-AOS767 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Solution of linear ill-posed problems using random dictionaries

In the present paper we consider application of overcomplete dictionaries to solution of general ill-posed linear inverse problems. In the context of regression problems, there has been enormous amount of effort to recover an unknown function using such dictionaries. One of the most popular methods, lasso and its versions, is based on minimizing empirical likelihood and unfortunately, requires stringent assumptions on the dictionary, the, so called, compatibility conditions. Though compatibility conditions are hard to satisfy, it is well known that this can be accomplished by using random dictionaries. In the present paper, we show how one can apply random dictionaries to solution of ill-posed linear inverse problems. We put a theoretical foundation under the suggested methodology and study its performance via simulations

Functional deconvolution in a periodic setting: Uniform case

We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model. We derive minimax lower bounds for the $L^2$-risk in the proposed functional deconvolution model when $f(\cdot)$ is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of $f(\cdot)$ that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls. In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings.Comment: Published in at http://dx.doi.org/10.1214/07-AOS552 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org