Random sampling and reconstruction of concentrated signals in a reproducing kernel space

TL;DR

This paper develops a probabilistic sampling and reconstruction method for signals concentrated on bounded domains within a metric measure space, establishing stability and providing an algorithm for approximate recovery from random samples.

Contribution

It introduces a weighted stability analysis for random sampling schemes in reproducing kernel spaces and proposes an algorithm for approximate signal reconstruction from random samples.

Findings

High-probability approximate reconstruction from random samples

Sampling size proportional to measure times logarithm for reliable recovery

Performance demonstrated with varying sampling densities

Abstract

In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain $\Omega$ of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in $\Omega$. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in $\Omega$, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain $\Omega$. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain $\Omega$ can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on $\Omega$, provided that the sampling size is at least of the order $\mu(\Omega) \ln (\mu(\Omega))$, where $\mu(\Omega)$ is the measure of the concentrated domain $\Omega$. Finally, we demonstrate the performance of proposed approximations to the original concentrated signal when the sampling procedure is taken either with small density or randomly with large size.