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Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

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Abstract

In compressed sensing, we seek to gain information about a vector x∈ℝN from d N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, 2006; Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006; Donoho, IEEE Trans. Inf. Theory 52:1289–1306, 2006) proposed to seek a good approximation to x via 1 minimization. In this paper, we show that in the case of Gaussian measurements, 1 minimization recovers the signal well from inaccurate measurements, thus improving the result from Candes et al. (Commun. Pure Appl. Math. 59:1207–1223, 2006). We also show that this numerically friendly algorithm (see Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006) with overwhelming probability recovers the signal with accuracy, comparable to the accuracy of the best k-term approximation in the Euclidean norm when kd/ln N.

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Correspondence to P. Wojtaszczyk.

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Communicated by Emmanuel Candes.

P. Wojtaszczyk would like to express his gratitude to Piotr Mankiewicz for answering questions about convex geometry and to Albert Cohen, Wolfgang Dahmen, and Ron DeVore for teaching him compressed sensing. Thanks are also due to Rachel Ward for an interesting discussion about stability results. The presentation of this paper was greatly improved by numerous and most helpful remarks of anonymous referees to whom the author would like to express his sincere thanks. This research was made possible by EC Marie Curie ToK program SPADE-2 at IMPAN. It was partially supported by MNiSW grant no. N201 269335.

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Wojtaszczyk, P. Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing. Found Comput Math 10, 1–13 (2010). https://doi.org/10.1007/s10208-009-9046-4

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  • DOI: https://doi.org/10.1007/s10208-009-9046-4

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