AI 中文总结
该研究针对内存AMP（MAMP）相关算法，解决GD-MAMP的溢出问题，提出两种低复杂度变体，推导通用梯度公式及WS-CG-VAMP(r)，对比不同算法的收敛性能。
AI 中文摘要
近似消息传递（AMP）类算法被广泛应用于高维噪声线性系统中的信号恢复。最近，名为内存AMP（MAMP）的框架被提出，为在AMP算法中引入内存项提供了新方法。在此基础上，针对右酉不变矩阵，研究人员提出了低复杂度梯度下降MAMP（GD-MAMP）。本文首先解决了GD-MAMP中由中间变量超出浮点范围导致的溢出问题，该问题通常在条件数较大时出现。其次，研究人员提出了GD-MAMP的两种低复杂度变体：一种用部分内存替代全长内存，另一种将每次迭代的矩阵-向量乘积数量减少1/3（从3次降至2次），且两种变体均未明显降低收敛速度。第三，研究人员开发了用于设计MAMP算法的通用梯度公式，该公式可将带热启动的共轭梯度VAMP（WS-CG-VAMP）作为特例包含在内。此外，研究人员表明该公式中正交化参数的计算可能会出现灾难性相消，这解释了WS-CG-VAMP的有限精度不稳定性。最后，研究人员推导了等效重构形式WS-CG-VAMP(r)，其矩阵-向量乘积数量最多可减少50%。以矩阵-向量乘积数量为衡量标准，GD-MAMP在条件数较小时收敛更快，而WS-CG-VAMP(r)在高精度算法下条件数较大时收敛更快，但由于灾难性相消，在IEEE双精度运算中可能会发散。
英文摘要
Approximate message passing (AMP)-type algorithms are widely used for signal recovery in high-dimensional noisy linear systems. Recently, a framework called memory AMP (MAMP) was introduced, offering a new approach to incorporating memory terms within AMP algorithms. Building on this, a low-complexity gradient descent MAMP (GD-MAMP) was proposed for right-unitarily invariant matrices. In this paper, we first address an overflow problem in GD-MAMP caused by intermediate variables exceeding the floating-point range, which typically occurs when the condition number is large. Second, we propose two low-complexity variants of GD-MAMP: one replaces full-length memory with partial memory, while the other reduces the number of matrix-vector products per iteration by $1/3$ (from three to two). Neither degrades the convergence speed notably. Third, we develop a general gradient-based formulation for designing MAMP algorithms. This formulation recovers warm-started conjugate gradient VAMP (WS-CG-VAMP) as a special case. Furthermore, we show that the computation of the orthogonalization parameters in this formulation can suffer from catastrophic cancellation, which explains the finite-precision instability of WS-CG-VAMP. Finally, we derive an equivalent reformulation, termed WS-CG-VAMP(r), which reduces the number of matrix-vector products by up to $50\%$. Measured by matrix-vector products, GD-MAMP converges faster for small condition numbers, whereas WS-CG-VAMP(r) converges faster for large ones under high-precision arithmetic but may diverge in IEEE double precision due to catastrophic cancellation.
