Mathematical Theory of Atomic Norm Denoising In Blind Two-Dimensional Super-Resolution (Extended Version)

TL;DR

This paper introduces a new mathematical framework for denoising and parameter estimation in blind 2D super-resolution using atomic norm minimization, with proven accuracy bounds and extensive simulations.

Contribution

It develops a novel theoretical approach for blind 2D super-resolution denoising and parameter estimation based on atomic norm, with explicit error bounds.

Findings

High-accuracy signal estimation under certain sample conditions

Explicit mean-squared error dependence on noise and system parameters

Validation of theoretical results through extensive simulations

Abstract

This paper develops a new mathematical framework for denoising in blind two-dimensional (2D) super-resolution upon using the atomic norm. The framework denoises a signal that consists of a weighted sum of an unknown number of time-delayed and frequency-shifted unknown waveforms from its noisy measurements. Moreover, the framework also provides an approach for estimating the unknown parameters in the signal. We prove that when the number of the observed samples satisfies certain lower bound that is a function of the system parameters, we can estimate the noise-free signal, with very high accuracy, upon solving a regularized least-squares atomic norm minimization problem. We derive the theoretical mean-squared error of the estimator, and we show that it depends on the noise variance, the number of unknown waveforms, the number of samples, and the dimension of the low-dimensional space where the unknown waveforms lie. Finally, we verify the theoretical findings of the paper by using extensive simulation experiments.