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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 &#x03B4; &gt; 0, we prove first that, given a Gaussian random frame of RN with M vectors, the worst-case &#x2113;2-error of consistent signal reconstruction decays with high probability as O((N/M) log(M/&#x221A;N)) uniformly for all signals of the unit ball BN &#x2282; RN. Up to a log factor, this matches a known lower bound in &#x03A9;(N/M) and former empirical validations in O(N/M). Equivalently, if M exceeds a minimal number of frame coefficients growing like O((N)/(&#x03B5;0) log(&#x221A;N)/(E0)), any vectors in BN with M identical quantized projections are at most E0 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 RN have a worst case error that decreases with high probability as O((K)/(M) log(MN)/(&#x221A;K3)) 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.

Error Decay of (Almost) Consistent Signal Estimations From Quantized Gaussian Random Projections