存储器AMP
Memory AMP
摘要：
近似消息传递（AMP）是一种低成本的迭代参数估计技术，用于某些高维非高斯分布的线性系统。AMP仅适用于独立同分布（IID）变换矩阵，但可能变得不可靠（例如，表现不佳或甚至发散）的其他矩阵合奏，特别是病态的。针对这一问题，提出了适用于一般右酉不变矩阵的正交/向量AMP（OAMP/VAMP）算法。然而，贝叶斯最优OAMP/VAMP（BO-OAMP/VAMP）需要高复杂度的线性最小均方误差（MMSE）估计器。这使得OAMP/VAMP无法在大规模系统中使用。针对AMP和BO-OAMP/VAMP算法的不足，提出了一种基于正交性原理的记忆AMP（MAMP）算法框架，保证了MAMP算法的估计误差渐近IID高斯分布。为了实现所需的正交MAMP，我们提供了一个正交化的本地记忆估计程序。此外，我们提出了一个贝叶斯最优MAMP（BO-MAMP），其中一个长记忆匹配滤波器用于干扰抑制。BO-MAMP的复杂性与AMP相当。为了渐近表征BO-MAMP的性能，推导出状态演化。基于状态演化方法对BO-MAMP中的弛豫参数和阻尼矢量进行了优化。最关键的是，对于所有右酉不变矩阵，优化后的BO-MAMP的状态演化收敛到与高复杂度的BO-OAMP/VAMP相同的不动点，并且如果其状态演化存在唯一不动点，则达到副本方法预测的贝叶斯最优均方误差.最后通过仿真验证了理论分析的正确性和有效性。
英文摘要：
Approximate message passing (AMP) is a low-cost iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions. AMP only applies to independent identically distributed (IID) transform matrices, but may become unreliable (e.g., perform poorly or even diverge) for other matrix ensembles, especially for ill-conditioned ones. To solve this issue, orthogonal/vector AMP (OAMP/VAMP) was proposed for general right-unitarily-invariant matrices. However, the Bayes-optimal OAMP/VAMP (BO-OAMP/VAMP) requires a high-complexity linear minimum mean square error (MMSE) estimator. This prevents OAMP/VAMP from being used in large-scale systems. To address the drawbacks of AMP and BO-OAMP/VAMP, this paper offers a memory AMP (MAMP) framework based on the orthogonality principle, which ensures that estimation errors in MAMP are asymptotically IID Gaussian. To realize the required orthogonality for MAMP, we provide an orthogonalization procedure for the local memory estimators. In addition, we propose a Bayes-optimal MAMP (BO-MAMP), in which a long-memory matched filter is used for interference suppression. The complexity of BO-MAMP is comparable to AMP. To asymptotically characterize the performance of BO-MAMP, a state evolution is derived. The relaxation parameters and damping vector in BO-MAMP are optimized based on state evolution. Most crucially, the state evolution of the optimized BO-MAMP converges to the same fixed point as that of the high-complexity BO-OAMP/VAMP for all right-unitarily-invariant matrices, and achieves the Bayes optimal MSE predicted by the replica method if its state evolution has a unique fixed point. Finally, simulations are provided to verify the theoretical results' validity and accuracy.
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