Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
 Emmanuel J. Candes,
 Justin Romberg
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Abstract
In this paper we develop a robust uncertainty principle for finite signals in ${\Bbb C}^N$ which states that, for nearly all choices $T, \Omega\subset\{0,\ldots,N1\}$ such that $T + \Omega \asymp (\log N)^{1/2} \cdot N,$ there is no signal $f$ supported on $T$ whose discrete Fourier transform $\hat{f}$ is supported on $\Omega.$ In fact, we can make the above uncertainty principle quantitative in the sense that if $f$ is supported on $T,$ then only a small percentage of the energy (less than half, say) of $\hat{f}$ is concentrated on $\Omega.$ As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids $f(s) = \sum_{t\in T}\alpha_1(t)\delta(st) + \sum_{\omega\in\Omega}\alpha_2(\omega)e^{i2\pi \omega s/N}/\sqrt{N}.$ We show that if a generic signal $f$ has a decomposition $(\alpha_1,\alpha_2)$ using spike and frequency locations in $T$ and $\Omega,$ respectively, and obeying $T + \Omega \leq {\rm Const}\cdot (\log N)^{1/2}\cdot N,$ then $(\alpha_1, \alpha_2)$ is the unique sparsest possible decomposition (all other decompositions have more nonzero terms). In addition, if $T + \Omega \leq {\rm Const}\cdot (\log N)^{1}\cdot N,$ then the sparsest $(\alpha_1,\alpha_2)$ can be found by solving a convex optimization problem. Underlying our results is a new probabilistic approach which insists on finding the correct uncertainty relation, or the optimally sparse solution for nearly all subsets but not necessarily all of them, and allows us to considerably sharpen previously known results [9], [10]. In fact, we show that the fraction of sets $(T, \Omega)$ for which the above properties do not hold can be upper bounded by quantities like $N^{\alpha}$ for large values of $\alpha.$ The QRUP (and the application to finding sparse representations) can be extended to general pairs of orthogonal bases $\Phi_1,\Phi_2 \mbox{of} {\Bbb C}^N.$ For nearly all choices ${\Gamma_1},{\Gamma_2}\subset\{0,\ldots,N1\}$ obeying ${\Gamma_1} + {\Gamma_2} \asymp \mu(\Phi_1,\Phi_2)^{2} \cdot (\log N)^{m},$ where $m\leq 6,$ there is no signal $f$ such that $\Phi_1 f$ is supported on ${\Gamma_1}$ and $\Phi_2 f$ is supported on ${\Gamma_2}$ where $\mu(\Phi_1,\Phi_2)$ is the mutual coherence between $\Phi_1$ and $\Phi_2.$
 Title
 Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
 Journal

Foundations of Computational Mathematics
Volume 6, Issue 2 , pp 227254
 Cover Date
 20060401
 DOI
 10.1007/s102080040162x
 Print ISSN
 16153375
 Online ISSN
 16153383
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Uncertainty principle
 Applications of uncertainty principles
 Random matrices
 Eigenvalues of random matrices
 Sparsity
 Trigonometric expansion
 Convex optimization
 Duality in optimization
 Basis pursuit
 Wavelets
 Linear programming
 Industry Sectors
 Authors

 Emmanuel J. Candes ^{(1)}
 Justin Romberg ^{(1)}
 Author Affiliations

 1. Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, USA