Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Introduction to Coding Theory
Quantization of Sparse Representations
DCC '07 Proceedings of the 2007 Data Compression Conference
Explicit constructions for compressed sensing of sparse signals
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
A Frame Construction and a Universal Distortion Bound for Sparse Representations
IEEE Transactions on Signal Processing
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
| Hi-index | 0.01 |
This paper studies the stability of some reconstruction algorithms for compressed sensing in terms of the bit precision. Considering the fact that practical digital systems deal with discretized signals, we motivate the importance of the total number of accurate bits needed from the measurement outcomes in addition to the number of measurements. It is shown that if one uses a 2k × n Vandermonde matrix with roots on the unit circle as the measurement matrix, O(l+k log n/k) bits of precision per measurement are sufficient to reconstruct a k-sparse signal x ơ Rn with dynamic range (i.e., the absolute ratio between the largest and the smallest nonzero coefficients) at most 2l within l bits of precision, hence identifying its correct support. Finally, we obtain an upper bound on the total number of required bits when the measurement matrix satisfies a restricted isometry property, which is in particular the case for random Fourier and Gaussian matrices. For very sparse signals, the upper bound on the number of required bits for Vandermonde matrices is shown to be better than this general upper bound.