New Analysis of Manifold Embeddings and Signal Recovery from Compressive Measurements

Armin Eftekhari, Michael B. Wakin

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

Compressive Sensing (CS) exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive, often random linear measurements of that signal. Strong theoretical guarantees have been established concerning the embedding of a sparse signal family under a random measurement operator and on the accuracy to which 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. Using tools from the theory of empirical processes, we improve upon previous results concerning the embedding of low-dimensional manifolds under random measurement operators. We also establish both deterministic and probabilistic instance-optimal bounds in ℓ2ℓ2 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 evidence that manifold-based models can be used with high accuracy in compressive signal processing.
Original languageEnglish
Pages (from-to)67-109
Number of pages43
JournalApplied and Computational Harmonic Analysis
Issue number1
Early online date6 Sept 2014
Publication statusPublished - 1 Jul 2015


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