关于无噪线性估计关于测量矩阵的普适性
On the Universality of Noiseless Linear Estimation with Respect to the Measurement Matrix
摘要：
在无噪声线性估计问题中，人们的目标是根据其线性投影y=Phi x* 的知识来重建向量x*。已经有许多理论工作集中在矩阵Phi是随机i.i.d.的情况下。一个，但一些启发性的证据表明，许多这些结果是普遍的，并远远超出了这一限制的情况下延伸。在这里，我们重新审视这个问题，通过棱镜的消息传递方法的发展，并考虑不仅是普遍性的l1过渡，如前所述，但也是一个最佳的贝叶斯重建。我们观察到，普遍性扩展到贝叶斯最优最小均方误差（MMSE），并在一定范围内的结构化矩阵。
英文摘要：
In a noiseless linear estimation problem, one aims to reconstruct a vector x* from the knowledge of its linear projections y=Phi x*. There have been many theoretical works concentrating on the case where the matrix Phi is a random i.i.d. one, but a number of heuristic evidence suggests that many of these results are universal and extend well beyond this restricted case. Here we revisit this problematic through the prism of development of message passing methods, and consider not only the universality of the l1 transition, as previously addressed, but also the one of the optimal Bayesian reconstruction. We observed that the universality extends to the Bayes-optimal minimum mean-squared (MMSE) error, and to a range of structured matrices.
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