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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
##### Keywords:
combinatorial optimization
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##### References:
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