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Abstract: Compressed Sensing CS seeks to recover an unknown vector with $N$ entriesby making far fewer than $N$ measurements; it posits that the number ofcompressed sensing measurements should be comparable to the information contentof the vector, not simply $N$. CS combines the important task of compressiondirectly with the measurement task. Since its introduction in 2004 there havebeen hundreds of manuscripts on CS, a large fraction of which developalgorithms to recover a signal from its compressed measurements. Because of theparadoxical nature of CS - exact reconstruction from seemingly undersampledmeasurements - it is crucial for acceptance of an algorithm that rigorousanalyses verify the degree of undersampling the algorithm permits. TheRestricted Isometry Property RIP has become the dominant tool used for theanalysis in such cases. We present here an asymmetric form of RIP which givestighter bounds than the usual symmetric one. We give the best known bounds onthe RIP constants for matrices from the Gaussian ensemble. Our derivationsillustrate the way in which the combinatorial nature of CS is controlled. Ourquantitative bounds on the RIP allow precise statements as to how aggressivelya signal can be undersampled, the essential question for practitioners. We alsodocument the extent to which RIP gives precise information about the trueperformance limits of CS, by comparing with approaches from high-dimensionalgeometry.



Author: Jeffrey D. Blanchard, Coralia Cartis, Jared Tanner

Source: https://arxiv.org/



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