Manifold-Based Signal Recovery and Parameter Estimation from Compressive Measurements

TL;DR

This paper extends compressive sensing to manifold models, providing theoretical guarantees for signal recovery and parameter estimation from noisy measurements, demonstrating high accuracy in high-dimensional data processing.

Contribution

It introduces deterministic and probabilistic bounds for manifold-based recovery, broadening the scope beyond sparse models in compressive sensing.

Findings

Probabilistic bounds outperform deterministic ones in accuracy.

Manifold models enable high-precision recovery from fewer measurements.

Empirical evidence supports the effectiveness of manifold-based compressive sensing.

Abstract

A field known as Compressive Sensing (CS) has recently emerged to help address the growing challenges of capturing and processing high-dimensional signals and data sets. CS exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive (or random) linear measurements of that signal. Strong theoretical guarantees have been established on the accuracy to which sparse or near-sparse signals can be recovered from noisy compressive measurements. In this paper, we address similar questions in the context of a different modeling framework. Instead of sparse models, we focus on the broad class of manifold models, which can arise in both parametric and non-parametric signal families. Building upon recent results concerning the stable embeddings of manifolds within the measurement space, we establish both deterministic and probabilistic instance-optimal bounds in $\ell_2$ for manifold-based signal recovery and parameter estimation from noisy compressive measurements. In line with analogous results for sparsity-based CS, we conclude that much stronger bounds are possible in the probabilistic setting. Our work supports the growing empirical evidence that manifold-based models can be used with high accuracy in compressive signal processing.