Approximate Message Passing With Consistent Parameter Estimation and Applications to Sparse Learning
Approximate Message Passing With Consistent Parameter Estimation and Applications to Sparse Learning
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DOI：
10.1109/tit.2014.2309005
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发表时间：
2012-07
影响因子：
2.5
通讯作者：
U. Kamilov;S. Rangan;A. Fletcher;M. Unser
中科院分区：
文献类型：
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作者：
U. Kamilov;S. Rangan;A. Fletcher;M. Unser
我们考虑一个独立同分布（i.i.d.）（可能是非高斯的）向量x ∈ Rn，其来自通过一般级联模型获得的测量y ∈ Rm，所述一般级联模型由已知线性变换和随后的概率分量（可能是非线性的）测量通道组成。提出了一种新的自适应广义近似消息传递方法。它使得能够联合学习先验信道和测量信道的统计数据沿着未知向量x的估计。我们证明，对于大的i.i.d.高斯变换矩阵，自适应GAMP的渐近分量的行为预测的一组简单的标量状态演化方程。此外，我们表明，自适应GAMP产生渐近一致的参数估计，当一定的最大似然估计可以在每一步进行。这意味着该算法实现了与知道正确参数值的oracle算法等效的重建质量。值得注意的是，这一结果适用于基本上任意参数化的未知分布，包括非线性和非高斯的。因此，自适应GAMP方法提供了一个系统的，一般的和计算效率高的方法，适用于大范围的线性非线性模型与可证明的保证。
We consider the estimation of an independent and identically distributed (i.i.d.) (possibly non-Gaussian) vector x ∈ Rn from measurements y ∈ Rm obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. A novel method, called adaptive generalized approximate message passing (adaptive GAMP) is presented. It enables the joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector x. We prove that, for large i.i.d. Gaussian transform matrices, the asymptotic componentwise behavior of the adaptive GAMP is predicted by a simple set of scalar state evolution equations. In addition, we show that the adaptive GAMP yields asymptotically consistent parameter estimates, when a certain maximum-likelihood estimation can be performed in each step. This implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. Remarkably, this result applies to essentially arbitrary parametrizations of the unknown distributions, including nonlinear and non-Gaussian ones. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of linear-nonlinear models with provable guarantees.