1. Error Decay of (Almost) Consistent Signal Estimations From Quantized Gaussian Random Projections.
- Author
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Jacques, Laurent
- Subjects
- *
COMPRESSED sensing , *SAMPLING errors , *SIGNAL quantization , *SCALAR field theory , *REMOTE sensing - Abstract
This paper provides new error bounds on consistent reconstruction methods for signals observed from quantized random projections. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a new observation of the estimated signal under the same sensing model. Focusing on dithered uniform scalar quantization of resolution \delta >0 , we prove first that, given a Gaussian random frame of \mathbb R^{N} with M vectors, the worst-case \ell 2 -error of consistent signal reconstruction decays with high probability as O(({N}/{M})\log ({M}/{\sqrt {N}})) uniformly for all signals of the unit ball \mathbb B^N \subset \mathbb R^N . Up to a log factor, this matches a known lower bound in \Omega (N/M) and former empirical validations in O(N/M) . Equivalently, if M exceeds a minimal number of frame coefficients growing like , any vectors in \mathbb B^{N} with M identical quantized projections are at most \epsilon 0 apart with high probability. Second, in the context of quantized compressed sensing with M Gaussian random measurements and under the same scalar quantization scheme, consistent reconstructions of K -sparse signals of \mathbb R^{N} have a worst case error that decreases with high probability as O(({K})/({M})\log ({MN})/({\sqrt {K}^{3}}))$ uniformly for all such signals. Finally, we show that the proximity of vectors whose quantized random projections are only approximately consistent can still be bounded with high probability. A certain level of corruption is thus allowed in the quantization process, up to the appearance of a systematic bias in the reconstruction error of (almost) consistent signal estimates. [ABSTRACT FROM PUBLISHER]
- Published
- 2016
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