Abstract We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the L p -risk ( 1 ≤ p ∞ ) . Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.
The case where @math for each @math ; (i.e., the case where in the multichannel deconvolution model the errors follow independent standard Brownian motions) was first considered in @cite_16 (extending the results obtained in @cite_2 for the case @math ).
@cite_16 @cite_2
2537174396 2884699236 2011360117 2963393250
This paper considers recovering @math -dimensional vectors @math , and @math from their circular convolutions @math . The vector @math is assumed to be @math -sparse in a known basis that is spread out in the Fourier domain, and each input @math is a member of a known @math -dimensional random subspace. We prove that whenever @math , the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey @math . By “diverse inputs,” we mean that the @math ’s belong to different, generic subspaces. To the best of our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which @math belongs is not fixed but needs to be determined. We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients and the delays in the channel impulse response @math are unknown. The encoder codes the @math -dimensional message vectors randomly and then transmits coded messages @math ’s over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than @math , and the number of messages is roughly at least @math . Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some @math errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are @math channels, which cause at most @math or @math errors, where @math t_ 1 @math pN @math 0 . If the information codeword belongs to a code with minimum distance @math , the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance @math that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions. We consider the problem of estimating the unknown response function in the multichannel deconvolution model with a boxcar-like kernel which is of particular interest in signal processing. It is known that, when the number of channels is finite, the precision of reconstruction of the response function increases as the number of channels @math grow (even when the total number of observations @math for all channels @math remains constant) and this requires that the parameter of the channels form a Badly Approximable @math -tuple. Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels @math grow with the total number of observations @math . However, in real-life situations, the number of channels @math usually refers to the number of physical devices and, consequently, may grow to infinity only at a slow rate as @math . When @math grows slowly as @math increases, we develop a procedure for the construction of a Badly Approximable @math -tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this @math -tuple, which explicitly shows its dependence on @math as @math is growing. This result is further used for the evaluation of the @math -risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels @math which minimizes the @math -risk. We consider the problem of estimating the unknown response function in the multichannel deconvolution model with long-range dependent Gaussian or sub-Gaussian errors. We do not limit our consideration to a specific type of long-range dependence rather we assume that the errors should satisfy a general assumption in terms of the smallest and largest eigenvalues of their covariance matrices. We derive minimax lower bounds for the quadratic risk in the proposed multichannel deconvolution model when the response function 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 the response function that is asymptotically optimal (in the minimax sense), or nearoptimal (within a logarithmic factor), in a wide range of Besov balls, for both Gaussian and sub-Gaussian errors. It is shown that the optimal convergence rates depend on the balance between the smoothness parameter of the response function, the kernel parameters of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the total number of channels. Some examples 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 are used to illustrate the application of the theory we developed. The optimal convergence rates and the adaptive estimators we consider extend the ones studied by Pensky and Sapatinas (2009, 2010) for independent and identically distributed Gaussian errors to the case of long-range dependent Gaussian or sub-Gaussian errors.