We consider the problem of recovering a complete (i.e., square and invertible) matrix $\mathbf A_0$, from $\mathbf Y \in \mathbb{R}^{n \times p}$ with $\mathbf Y = \mathbf A_0 \mathbf X_0$, provided $\mathbf X_0$ is sufficiently sparse. This recovery problem is central to the theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals, and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers $\mathbf A_0$ when $\mathbf X_0$ has $O(n)$ nonzeros per column, under suitable probability model for $\mathbf X_0$. In contrast, prior results based on efficient algorithms provide recovery guarantees when $\mathbf X_0$ has only $O(n^{1-\delta})$ nonzeros per column for any constant $\delta \in (0, 1)$. Our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. To show this apparently hard problem is tractable, we first provide a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no "spurious" local minimizers. This particular geometric structure allows us to design a Riemannian trust region algorithm over the sphere that provably converges to a local minimizer with an arbitrary initialization, despite the presence of saddle points. The geometric approach we develop here may also shed light on other problems arising from nonconvex recovery of structured signals. This paper provides an overview of key algorithmic components and an quantitative characterization of the objective landscape. In the companion paper (to be linked), the whole algorithmic pipeline and rigorous performance guarantees are presented.

研究矩阵恢复的问题，探讨了通过优化解决该问题的方法，证明了当稀疏度为$O(n)$时，算法具有恢复原始矩阵的能力，并提供了实现该方法的最新算法。

球面上的完整字典恢复I：概述和几何图像