相关论文: A Probabilistic Oracle Inequality and Quantificati…
We tackle the problem of estimating a regression function observed in an instrumental regression framework. This model is an inverse problem with unknown operator. We provide a spectral cut-off estimation procedure which enables to derive…
In this paper,we consider a high-dimensional statistical estimation problem in which the the number of parameters is comparable or larger than the sample size. We present a unified analysis of the performance guarantees of exponential…
This paper deals with recovering an unknown vector $\theta$ from the noisy data $Y=A\theta+\sigma\xi$, where $A$ is a known $(m\times n)$-matrix and $\xi$ is a white Gaussian noise. It is assumed that $n$ is large and $A$ may be severely…
We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension $D$. Since calculating the singular value decomposition (SVD) only for the largest singular values is much…
This paper considers the problem of estimating a periodic function in a continuous time regression model with a general square integrable semimartingale noise. A model selection adaptive procedure is proposed. Sharp non-asymptotic oracle…
Many statistical estimation procedures lead to nonconvex optimization problems. Algorithms to solve these are often guaranteed to output a stationary point of the optimization problem. Oracle inequalities are an important theoretical…
Convergence rates in spectral regularization methods quantify the approximation error in inverse problems as a function of the noise level or the number of sampling points. Classical strong convergence rate results typically rely on source…
A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. However, these so-called filter methods are generally restricted to monotonic transformations,…
For linear inverse problems $Y=\mathsf{A}\mu+\xi$, it is classical to recover the unknown signal $\mu$ by iterative regularisation methods $(\widehat \mu^{(m)}, m=0,1,\ldots)$ and halt at a data-dependent iteration $\tau$ using some…
The authors study statistical linear inverse problems in Hilbert spaces. Approximate solutions are sought within a class of linear one-parameter regularization schemes, and the parameter choice is crucial to control the root mean squared…
Further development of the method of computational experiments for solving ill-posed problems is given. The effective (unoverstated) estimate for solution error of the first-kind equation is obtained using the truncating singular numbers…
The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between…
We study the problem of estimating the one-point specification probabilities in non-necessary finite discrete random fields from partially observed independent samples. Our procedures are based on model selection by minimization of a…
We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence…
Motivated by emerging applications in machine learning, we consider an optimization problem in a general form where the gradient of the objective function is available through a biased stochastic oracle. We assume a bias-control parameter…
An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle…
We consider a general statistical linear inverse problem, where the solution is represented via a known (possibly overcomplete) dictionary that allows its sparse representation. We propose two different approaches. A model selection…
A general many quantiles + noise model is studied in the robust formulation (allowing non-normal, non-independent observations), where the identifiability requirement for the noise is formulated in terms of quantiles rather than the…
We investigate the construction of early stopping rules in the nonparametric regression problem where iterative learning algorithms are used and the optimal iteration number is unknown. More precisely, we study the discrepancy principle, as…
Recent algebraic parametric estimation techniques led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal. In this paper, we extend such differentiation methods by providing a larger choice…