In the problem of structured signal recovery from high-dimensional linear observations, it is commonly assumed that full-precision measurements are available. Under this assumption, the recovery performance of the popular Generalized Lasso (G-Lasso) is by now well-established. In this paper, we extend these types of results to the practically relevant settings with quantized measurements. We study two extremes of the quantization schemes, namely, uniform and one-bit quantization; the former imposes no limit on the number of quantization bits, while the second only allows for one bit. In the presence of a uniform dithering signal and when measurement vectors are sub-gaussian, we show that the same algorithm (i.e., the G-Lasso) has favorable recovery guarantees for both uniform and one-bit quantization schemes. Our theoretical results, shed light on the appropriate choice of the range of values of the dithering signal and accurately capture the error dependence on the problem parameters. For example, our error analysis shows that the G-Lasso with one-bit uniformly dithered measurements leads to only a logarithmic rate loss compared to the full-precision measurements.

中文翻译：

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具有抖动量化的亚高斯测量的广义套索

在从高维线性观测中恢复结构化信号的问题中，通常假设全精度测量是可用的。在这个假设下，流行的广义套索（G-Lasso）的恢复性能现在已经确立。在本文中，我们将这些类型的结果扩展到具有量化测量的实际相关设置。我们研究了量化方案的两个极端，即均匀量化和一位量化；前者对量化位数没有限制，而后者只允许一位。在存在均匀抖动信号且测量向量为亚高斯时，我们表明相同的算法（即 G-Lasso）对于均匀和一位量化方案都具有良好的恢复保证。我们的理论结果，shed light on the appropriate choice of the range of values of the dithering signal and accurately capture the error dependence on the problem parameters. 例如，我们的误差分析表明，与全精度测量相比，具有一位均匀抖动测量的 G-Lasso 仅导致对数速率损失。