On the Effective Measure of Dimension in the Analysis Cosparse Model

Raja Giryes*, Yaniv Plan, Roman Vershynin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

Many applications have benefited remarkably from low-dimensional models in the recent decade. The fact that many signals, though high dimensional, are intrinsically low dimensional has given the possibility to recover them stably from a relatively small number of their measurements. For example, in compressed sensing with the standard (synthesis) sparsity prior and in matrix completion, the number of measurements needed is proportional (up to a logarithmic factor) to the signal's manifold dimension. Recently, a new natural low-dimensional signal model has been proposed: the cosparse analysis prior. In the noiseless case, it is possible to recover signals from this model, using a combinatorial search, from a number of measurements proportional to the signal's manifold dimension. However, if we ask for stability to noise or an efficient (polynomial complexity) solver, all the existing results demand a number of measurements, which is far removed from the manifold dimension, sometimes far greater. Thus, it is natural to ask whether this gap is a deficiency of the theory and the solvers, or if there exists a real barrier in recovering the cosparse signals by relying only on their manifold dimension. Is there an algorithm which, in the presence of noise, can accurately recover a cosparse signal from a number of measurements proportional to the manifold dimension? In this paper, we prove that there is no such algorithm. Furthermore, we show through the numerical simulations that even in the noiseless case convex relaxations fail when the number of measurements is comparable with the manifold dimension. This gives a practical counterexample to the growing literature on the compressed acquisition of signals based on manifold dimension.

Original languageEnglish
Article number7185448
Pages (from-to)5745-5753
Number of pages9
JournalIEEE Transactions on Information Theory
Volume61
Issue number10
DOIs
StatePublished - 1 Oct 2015
Externally publishedYes

Funding

FundersFunder number
Humboldt Research Foundation
National Science FoundationDMS 1265782
Office of Naval Research
Directorate for Mathematical and Physical Sciences
Division of Mathematical Sciences1103909, 1265782
Air Force Office of Scientific Research
Army Research Office
National Sleep Foundation
Alexander von Humboldt-Stiftung
U.S. Air ForceFA9550-14-1-0009

    Keywords

    • Compressed sensing
    • manifold dimension
    • sparse representations
    • the analysis model
    • total variation

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