关于具有不连续去噪器的近似消息传递的状态演化
On the State Evolution of Approximate Message Passing with Discontinuous Denoisers
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中文总结 AI 辅助
本文针对不连续去噪器的AMP，修正Onsager系数加入边界项，证明其遵循状态演化，仿真显示硬切片器AMP接近贝叶斯最优。
中文摘要 AI 辅助
我们考虑具有不连续去噪器的近似消息传递（AMP），例如硬切片器、量化器和硬阈值，对于这些去噪器，经典的状态演化（SE）分析不适用。首先表明，标准的Onsager系数（即去噪器逐点导数的平均值）遗漏了一个边界项，即跳跃大小乘以去噪器输入在跳跃处的密度的总和。然后，对于实独立同分布（i.i.d.）高斯矩阵、正噪声方差、固定迭代次数以及具有有限二阶矩的先验，我们将Lipschitz SE定理直接推广到预先固定的、除有限个跳跃外连续可微且导数有界的去噪器，通过限制AMP与在去噪器的Lipschitz斜坡上的AMP迭代之间的距离（利用跨越跳跃的输入比例），从而证明AMP遵循SE，对于符号错误等测试函数也是如此，前提是其Onsager系数包含边界项，该边界项要么沿SE评估，要么如我们所提出的，在从残差估计的噪声水平处评估，密度由已知先验计算。使用4级和8级脉冲幅度调制（PAM）的仿真结果表明，具有硬切片器的AMP（其标准系数恒为零）使用该系数会失败，而使用我们提出的系数，它遵循SE并接近贝叶斯最优AMP。
英文摘要
We consider approximate message passing (AMP) with discontinuous denoisers, such as hard slicers, quantizers and hard thresholding, for which the classical state evolution (SE) analysis does not apply. It is first shown that the standard Onsager coefficient; i.e., the average pointwise derivative of the denoiser, misses a boundary term, namely, the sum over the jumps of the jump size times the density of the denoiser input at the jump. Then, for real independent and identically distributed (i.i.d.) Gaussian matrices, a positive noise variance, a fixed number of iterations and a prior with finite second moment, we transfer the Lipschitz SE theorem directly to denoisers, fixed in advance, that are continuously differentiable with bounded derivative except at finitely many jumps, by bounding the distance between the iterates of AMP and of AMP on Lipschitz ramps of the denoisers through the fraction of inputs that cross a jump. It is thereby proven that AMP follows SE, also for test functions such as symbol errors, if its Onsager coefficient contains the boundary term, evaluated either along SE or, as we propose, at the noise level estimated from the residual, with the density computed from the known prior. Simulation results with 4- and 8-level pulse amplitude modulation (PAM) demonstrate that AMP with a hard slicer, whose standard coefficient is identically zero, fails with this coefficient, while with the proposed one, it follows SE and comes close to Bayes-optimal AMP.
发表机构
- School of Computer Science and Engineering, Constructor University（Constructor大学计算机科学与工程学院）
- Graduate School of Engineering, Osaka University（大阪大学研究生院工学研究科）
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