We consider the problem of recovering signals from their power spectral
density. This is a classical problem referred to in literature as the phase
retrieval problem, and is of paramount importance in many fields of applied
sciences. In general, additional prior information about the signal is required
to guarantee unique recovery as the mapping from signals to power spectral
density is not one-to-one. In this paper, we assume that the underlying signals
are sparse. Recently, semidefinite programming (SDP) based approaches were
explored by various researchers. Simulations of these algorithms strongly
suggest that signals upto $o(\sqrt{n})$ sparsity can be recovered by this
technique. In this work, we develop a tractable algorithm based on reweighted
$l_1$-minimization that recovers a sparse signal from its power spectral
density for significantly higher sparsities, which is unprecedented.
We discuss the square-root bottleneck of the existing convex algorithms and
show that a $k$-sparse signal can be efficiently recovered using $O(k^2logn)$
phaseless Fourier measurements. We also show that a $k$-sparse signal can be
recovered using only $O(k log n)$ phaseless measurements if we are allowed to
design the measurement matrices.

文章研究了利用 SFFT 算法去恢复信号在功率谱上的密度，此算法可以有效恢复高于 o (sqrt (n)) 的稀疏信号，并通过实验证明，其相比已有的波凸算法有着更出色的性能。

稀疏相位恢复：凸算法与限制

Sparse Phase Retrieval: Convex Algorithms and Limitations

L1-minimization refers to finding the minimum L1-norm solution to an
underdetermined linear system b=Ax. Under certain conditions as described in
compressive sensing theory, the minimum L1-norm solution is also the sparsest
solution. In this paper, our study addresses the speed and scalability of its
algorithms. In particular, we focus on the numerical implementation of a
sparsity-based classification framework in robust face recognition, where
sparse representation is sought to recover human identities from very
high-dimensional facial images that may be corrupted by illumination, facial
disguise, and pose variation. Although the underlying numerical problem is a
linear program, traditional algorithms are known to suffer poor scalability for
large-scale applications. We investigate a new solution based on a classical
convex optimization framework, known as Augmented Lagrangian Methods (ALM). The
new convex solvers provide a viable solution to real-world, time-critical
applications such as face recognition. We conduct extensive experiments to
validate and compare the performance of the ALM algorithms against several
popular L1-minimization solvers, including interior-point method, Homotopy,
FISTA, SESOP-PCD, approximate message passing (AMP) and TFOCS. To aid peer
evaluation, the code for all the algorithms has been made publicly available.

本研究探究基于凸优化框架的新解法（即增广拉格朗日方法）以提高 L1-minimization 在稀疏表示和面部识别方面的速度和可伸缩性。我们进行了大量实验并将算法代码公开以供同行评估。

快速 L1 最小化算法用于强韧人脸识别

Fast L1-Minimization Algorithms For Robust Face Recognition

We analyze an Iteratively Re-weighted Least Squares (IRLS) algorithm for
promoting l1-minimization in sparse and compressible vector recovery. We prove
its convergence and we estimate its local rate. We show how the algorithm can
be modified in order to promote lt-minimization for t<1, and how this
modification produces superlinear rates of convergence.

本文研究使用迭代加权最小二乘算法（IRLS）促进稀疏和可压缩向量恢复中的 l1 最小化，证明其收敛性和估计局部速率，并且展示了如何修改算法，以便在 t 小于 1 时促进 lt 最小化，并且这种修改有着超线性的收敛速率。