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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]

This paper establishes a nearly optimal algorithm for denoising a mixture of sinusoids from noisy equispaced samples. We derive our algorithm by viewing line spectral estimation as a sparse recovery problem with a continuous, infinite dictionary. We show how to compute the estimator via semidefinite programming and provide guarantees on its mean-squared error rate. We derive a complementary minimax lower bound on this estimation rate, demonstrating that our approach nearly achieves the best possible estimation error. Furthermore, we establish bounds on how well our estimator localizes the frequencies in the signal, showing that the localization error tends to zero as the number of samples grows. We verify our theoretical results in an array of numerical experiments, demonstrating that the semidefinite programming approach outperforms three classical spectral estimation techniques. [ABSTRACT FROM PUBLISHER]