The Generalized Lasso for Sub-gaussian Measurements with Dithered Quantization

TL;DR

This paper extends the analysis of the Generalized Lasso to settings with quantized measurements, demonstrating that it maintains favorable recovery guarantees under uniform and one-bit quantization with dithering.

Contribution

It provides theoretical guarantees for the G-Lasso in quantized measurement scenarios, including one-bit quantization, with insights on dithering and error dependence.

Findings

G-Lasso achieves near-optimal recovery with uniform dithering and sub-gaussian measurements.

One-bit quantized measurements with dithering cause only a logarithmic error rate loss.

Theoretical bounds guide the choice of dithering range for effective signal recovery.

Abstract

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.