The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine learning. However, in contrast to the classical sparse recovery problem, the most natural formulation for finding the sparsest vector in a subspace is usually nonconvex. In this paper, we overview recent advances on global nonconvex optimization theory for solving this problem, ranging from geometric analysis of its optimization landscapes, to efficient optimization algorithms for solving the associated nonconvex optimization problem, to applications in machine intelligence, representation learning, and imaging sciences. Finally, we conclude this review by pointing out several interesting open problems for future research.

中文翻译：

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在子空间中寻找最稀疏的向量：理论、算法和应用

在低维子空间中寻找最稀疏向量（方向）的问题可以被认为是稀疏恢复问题的齐次变体，它在鲁棒子空间恢复、字典学习、稀疏盲反卷积和信号处理中的许多其他问题中找到了应用和机器学习。然而，与经典的稀疏恢复问题相反，在子空间中寻找最稀疏向量的最自然的公式通常是非凸的。在本文中，我们概述了解决此问题的全局非凸优化理论的最新进展，从其优化景观的几何分析到用于解决相关非凸优化问题的有效优化算法，再到机器智能、表示学习和成像中的应用。科学。最后，