Multichannel Boxcar Deconvolution with Growing Number of Channels

TL;DR

This paper studies the estimation of an unknown response function in multichannel deconvolution with a boxcar kernel, focusing on scenarios where the number of channels grows slowly with the total observations, and develops methods for constructing badly approximable tuples to optimize estimation accuracy.

Contribution

It introduces a procedure for constructing badly approximable tuples when the number of channels grows slowly, and evaluates the impact on the $L^2$-risk of wavelet-based estimators.

Findings

Constructed badly approximable $M$-tuples with explicit bounds.

Derived $L^2$-risk bounds for the estimator.

Provided a method to choose the optimal number of channels.

Abstract

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 $M$ grow (even when the total number of observations $n$ for all channels $M$ remains constant) and this requires that the parameter of the channels form a Badly Approximable $M$-tuple. Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels $M=M_n$ grow with the total number of observations $n$. However, in real-life situations, the number of channels $M = M_n$ usually refers to the number of physical devices and, consequently, may grow to infinity only at a slow rate as $n \rightarrow \infty$. When $M=M_n$ grows slowly as $n$ increases, we develop a procedure for the construction of a Badly Approximable $M$-tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this $M$-tuple, which explicitly shows its dependence on $M$ as $M$ is growing. This result is further used for the evaluation of the $L^2$-risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels $M$ which minimizes the $L^2$-risk.