This work is concerned with the problem of recovering high-dimensional signals, which belong to a convex set of lowcomplexity, from a small number of quantized measurements. We propose to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one-bit and uniform multi-bit quantization. Assuming that the linear measurement process can be modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both schemes near-optimal uniform reconstruction guarantees by adding well-designed noise to the linear measurements prior to the quantization step. In the one-bit case, we show that the program is robust against adversarial bit corruptions as well as additive noise on the linear measurements. Further, our analysis quantifies precisely how the rate-distortion relationship of the program changes depending on whether we seek reconstruction accuracies above or below the noise floor. The proofs rely on recent results by Dirksen and Mendelson on non-Gaussian hyperplane tessellations.
Abstract-Optimal error probabilities for transmission over the average-power-limited Gaussian channel with intermittent feedback are studied. For the two-message case, the asymptotic decay of the probability of error in the blocklength is doubleexponential and is fully characterized. For positive rates a critical rate is identified below which a double-exponential decay is possible and above which it is not.
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