Iterative thresholding algorithms are well-suited for high-dimensional
problems in sparse recovery and compressive sensing. The performance of this
class of algorithms depends heavily on the tuning of certain threshold
parameters. In particular, both the final reconstruction error and the
convergence rate of the algorithm crucially rely on how the threshold parameter
is set at each step of the algorithm. In this paper, we propose a
parameter-free approximate message passing (AMP) algorithm that sets the
threshold parameter at each iteration in a fully automatic way without either
having an information about the signal to be reconstructed or needing any
tuning from the user. We show that the proposed method attains both the minimum
reconstruction error and the highest convergence rate. Our method is based on
applying the Stein unbiased risk estimate (SURE) along with a modified gradient
descent to find the optimal threshold in each iteration. Motivated by the
connections between AMP and LASSO, it could be employed to find the solution of
the LASSO for the optimal regularization parameter. To the best of our
knowledge, this is the first work concerning parameter tuning that obtains the
fastest convergence rate with theoretical guarantees.

本文提出一种无需用户调整阈值参数的自动方式，采用修改过的梯度下降算法和 Stein 无偏风险估计，实现高稀疏恢复和压缩感知算法中的最小重构误差和最高收敛速率，为参数调整领域的最快收敛率理论保障提供了解决方案。