Asymptotic Performance of Discrete-Valued Vector Reconstruction via Box-Constrained Optimization With Sum of $\ell _{1}$ Regularizers

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 ℓ
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.