In this paper, we analyze the asymptotic performance of convex optimization-based discrete-valued vector reconstruction from linear measurements. We firstly propose a box-constrained version of the conventional sum of absolute values (SOAV) optimization, which uses a weighted sum of $\ell _{1}$ regularizers as a regularizer for the discrete-valued vector. We then derive the asymptotic symbol error rate (SER) performance of the box-constrained SOAV (Box-SOAV) optimization theoretically by using the convex Gaussian min-max theorem (CGMT). We also derive the asymptotic distribution of the estimate obtained by the Box-SOAV optimization. On the basis of the asymptotic results, we can obtain the optimal parameters of the Box-SOAV optimization in terms of the asymptotic SER. Moreover, we can also optimize the quantizer to obtain the final estimate of the unknown discrete-valued vector. Simulation results show that the empirical SER performance of Box-SOAV and the conventional SOAV is very close to the theoretical result for Box-SOAV when the problem size is sufficiently large. We also show that we can obtain better SER performance by using the proposed asymptotically optimal parameters and quantizers compared to the case with some fixed parameter and a naive quantizer.

中文翻译：

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通过带$l_1$正则化器和的框约束优化离散值向量重构的渐近性能

在本文中，我们分析了基于凸优化的线性测量离散值向量重建的渐近性能。我们首先提出了传统绝对值总和 (SOAV) 优化的框约束版本，它使用 $\ell_{1}$ 正则化器的加权和作为离散值向量的正则化器。然后我们通过使用凸高斯最小-最大定理 (CGMT) 从理论上推导出框约束 SOAV (Box-SOAV) 优化的渐近符号错误率 (SER) 性能。我们还推导出 Box-SOAV 优化获得的估计值的渐近分布。在渐近结果的基础上，我们可以根据渐近SER得到Box-SOAV优化的最优参数。而且，我们还可以优化量化器以获得未知离散值向量的最终估计。仿真结果表明，当问题规模足够大时，Box-SOAV 和常规 SOAV 的经验 SER 性能非常接近 Box-SOAV 的理论结果。我们还表明，与使用一些固定参数和朴素量化器的情况相比，我们可以通过使用所提出的渐近最优参数和量化器获得更好的 SER 性能。