In \cite{Stojnicclupint19,Stojnicclupcmpl19,Stojnicclupplt19} we introduced CLuP, a \bl{\textbf{Random Duality Theory (RDT)}} based algorithmic mechanism that can be used for solving hard optimization problems. Due to their introductory nature, \cite{Stojnicclupint19,Stojnicclupcmpl19,Stojnicclupplt19} discuss the most fundamental CLuP concepts. On the other hand, in our companion paper \cite{Stojniccluplargesc20} we started the story of going into a bit deeper details that relate to many of other remarkable CLuP properties with some of them reaching well beyond the basic fundamentals. Namely, \cite{Stojniccluplargesc20} discusses how a somewhat silent RDT feature (its algorithmic power) can be utilized to ensure that CLuP can be run on very large problem instances as well. In particular, applying CLuP to the famous MIMO ML detection problem we showed in \cite{Stojniccluplargesc20} that its a large scale variant, $\text{CLuP}^{r_0}$, can handle with ease problems with \textbf{\emph{several thousands}} of unknowns with theoretically minimal complexity per iteration (only a single matrix-vector multiplication suffices). In this paper we revisit MIMO ML detection and discuss another remarkable phenomenon that emerges within the CLuP structure, namely the so-called \bl{\textbf{\emph{rephasing}}}. As MIMO ML enters the so-called low $\alpha$ regime (fat system matrix with ratio of the number of rows and columns, $\alpha$, going well below $1$) it becomes increasingly difficult even for the basic standard CLuP to handle it. However, the discovery of the rephasing ensures that CLuP remains on track and preserves its ability to achieve the ML performance. To demonstrate the power of the rephasing we also conducted quite a few numerical experiments, compared the results we obtained through them to the theoretical predictions, and observed an excellent agreement.

中文翻译：

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重定阶段的CLuP

在\ cite {Stojnicclupint19，Stojnicclupcmpl19，Stojnicclupplt19}中，我们介绍了CLuP，这是一种基于\ bl {\ textbf {Random Duality Theory（RDT）}}的算法机制，可用于解决硬优化问题。由于其介绍性，\ cite {Stojnicclupint19，Stojnicclupcmpl19，Stojnicclupplt19}讨论了最基本的CLuP概念。另一方面，在我们的配套文件\ cite {Stojniccluplargesc20}中，我们开始了一个涉及更深层细节的故事，这些细节与许多其他出色的CLuP特性有关，其中一些特性远远超出了基本原理。即，\ cite {Stojniccluplargesc20}讨论了如何利用某种无声的RDT功能（其算法功能）来确保CLuP也可以在非常大的问题实例上运行。尤其是，将CLuP应用于我们在\ cite {Stojniccluplargesc20}中展示的著名MIMO ML检测问题，它的大规模变体$ \ text {CLuP} ^ {r_0} $可以轻松解决\ textbf {\ emph { }}每次迭代理论上具有最小的复杂性的未知数（仅单个矩阵向量乘法就足够了）。在本文中，我们将重新探讨MIMO ML检测，并讨论在CLuP结构内出现的另一个显着现象，即所谓的\ bl {\ textbf {\ emph {rephasing}}}。随着MIMO ML进入所谓的低$ \ alpha $机制（脂肪系统矩阵与行和列数之比，$ \ alpha $，远低于$ 1 $），即使对于基本标准CLuP来说，也变得越来越困难。处理它。然而，重新定相的发现可确保CLuP保持在原样，并保持其实现ML性能的能力。为了证明重定相的力量，我们还进行了许多数值实验，将通过它们获得的结果与理论预测进行了比较，并观察到了极好的一致性。