Error Decay of (almost) Consistent Signal Estimations from Quantized Random Gaussian Projections

This paper provides new error bounds on consistent reconstruction methods for signals observed from quantized random sensing. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a reobservation of the estimated signal under the same sensing model. Focusing on dithered uniform scalar quantization of resolution δ > 0, we prove first that, given a random Gaussian frame of R with M vectors, the worst case `2-error of consistent signal reconstruction decays with high probability as O( M log M √ N ) uniformly for all signals of the unit ball B ⊂ R . Up to a log factor, this matches a known lower bound in Ω(N/M). Equivalently, with a minimal number of frame coefficients behaving like M = O( 0 log( √ N 0 )), any vectors in B with M identical quantized projections are at most 0 apart with high probability. Second, in the context of Quantized Compressed Sensing with M random Gaussian measurements and under the same scalar quantization scheme, consistent reconstructions of K-sparse signals of R have a worst case error that decreases with high probability as O( M log MN √ K 3 ) uniformly for all such signals. Finally, we show that the strict consistency condition can be slightly relaxed, e.g., allowing for a bounded level of error in the quantization process, while still guaranteeing a proximity between the original and the estimated signal. In particular, if this quantization error is of order O(1) with respect to M , similar worst case error decays are reached for reconstruction methods adjusted to such an approximate consistency.