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For most large underdetermined systems of equations, the minimal \(\ell_1\)-norm near-solution approximates the sparsest near-solution. (English) Zbl 1105.90068
Summary: We consider inexact linear equations \(y\approx\Phi x\) where \(y\) is a given vector in \(\mathbb{R}^n\), \(\Phi\) is a given \(n\times m\) matrix, and we wish to find \(x_{0,\varepsilon}\) as sparse as possible while obeying \(\|y-\Phi x_{0, \varepsilon}\}_2\leq\varepsilon\). In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the \(\ell_1\)-minimization problem \[ \min\|x\|_1\quad\text{ subject to }\quad\|y-\Phi x\|_2 \leq\varepsilon \] is convex and is considered tractable. We show that for most \(\Phi\), if the optimally sparse approximation \(x_{0,\varepsilon}\) is sufficiently sparse, then the solution \(x_{1,\varepsilon}\) of the \(\ell_1\)-minimization problem is a good approximation to \(x_{0,\varepsilon}\). We suppose that the columns of \(\Phi\) are normalized to the unit \(\ell_2\)-norm, and we place uniform measure on such \(\Phi\). We study the underdetermined case where \(m\sim\tau n\) and \(\tau>1\), and prove the existence of \(\rho=\rho (\tau)>0\) and \(C=C(\rho,\tau)\) so that for large \(n\) and for all \(\Phi\)’s except a negligible fraction, the following approximate sparse solution property of \(\Phi\) holds: for every \(y\) having an approximation \(\|y-\Phi x_0\|_2\leq\varepsilon\) by a coefficient vector \(x_0\in\mathbb{R}^m\) with fewer than \(\rho\cdot n\) nonzeros, \[ \|x_{1,\varepsilon}-x_0\|_2\leq C \cdot\varepsilon. \] This has two implications. First, for most \(\Phi\), whenever the combinatorial optimization result \(x_{0, \varepsilon}\) would be very sparse, \(x_{1,\varepsilon}\) is a good approximation to \(x_{0,\varepsilon}\). Second, suppose we are given noisy data obeying \(y=\Phi x_0+z\) where the unknown \(x_0\) is known to be sparse and the noise \(\|z\|\leq \varepsilon\). For most \(\Phi\), noise-tolerant \(\ell_1\)-minimization will stably recover \(x_0\) from \(y\) in the presence of noise \(z\). We also study the barely determined case \(m=n\) and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost-spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices.

MSC:
90C27 Combinatorial optimization
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References:
[1] Candès, IEEE Trans Inform Theory
[2] Chen, SIAM J Sci Comput 20 pp 33– (1998)
[3] ; ; ; Signal processing and compression with wavelet packets. Progress in wavelet analysis and applications (Toulouse, 1992), 77–93. Frontières, Gif-sur-Yvette, 1993.
[4] ; Local operator theory, random matrices and Banach spaces. Handbook of the geometry of Banach spaces, Vol. 1, 317–366. North-Holland, Amsterdam, 2001. · Zbl 1067.46008
[5] Donoho, Comm Pure Appl Math
[6] Donoho, Proc Natl Acad Sci USA 100 pp 2197– (2003)
[7] Donoho, IEEE Trans Inform Theory 52 pp 6– (2006)
[8] Donoho, IEEE Trans Inform Theory 47 pp 2845– (2001)
[9] Donoho, J Roy Statist Soc Ser B 54 pp 41– (1992)
[10] Some results on convex bodies and Banach spaces. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), 123–160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961.
[11] Edelman, SIAM J Matrix Anal Appl 9 pp 543– (1988)
[12] Efron, Ann Statist 32 pp 407– (2004)
[13] New results about random covariance matrices and statistical applications. Doctoral dissertation, Stanford University, 2004.
[14] Elad, IEEE Trans Inform Theory 48 pp 2558– (2002)
[15] Figiel, Acta Math 139 pp 53– (1977)
[16] Fuchs, IEEE Trans Inform Theory
[17] Fuchs, IEEE Trans Inform Theory 50 pp 1341– (2004)
[18] ; Matrix computations. Johns Hopkins, Baltimore, 1989.
[19] Gribonval, IEEE Trans Inform Theory 49 pp 3320– (2003)
[20] Kashin, Izv Akad Nauk SSSR Ser Mat 41 pp 334– (1977)
[21] The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, R.I., 2001.
[22] Mallat, IEEE Trans Signal Process 41 pp 3397– (1993)
[23] ; Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics, 1200. Springer, Berlin, 1986.
[24] Natarajan, SIAM J Comput 24 pp 227– (1995)
[25] The symmetric eigenvalue problem. Prentice-Hall Series in Computational Mathematics. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980.
[26] The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989. · doi:10.1017/CBO9780511662454
[27] Szarek, Bull Acad Polon Sci Sér Sci Math Astronom Phys 26 pp 691– (1978)
[28] Szarek, Amer J Math 112 pp 899– (1990)
[29] Szarek, J Complexity 7 pp 131– (1991)
[30] Tibshirani, J Roy Statist Soc Ser B 58 pp 267– (1996)
[31] Tropp, IEEE Trans Inform Theory 50 pp 2231– (2004)
[32] Tropp, IEEE Trans Inform Theory
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