We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.

翻译：我们研究回收一个部分观测的高层次矩阵的问题,该矩阵的柱体符合非线性结构,如子空间的组合、代数种类或组群。回收问题被表述为对原始矩阵适用的非线性地貌图的排位最小化,然后又被格拉斯曼方块的受限制的非线性优化问题进一步近似。我们提出了两套算法,其中一套来自里曼式优化,另一套是交替最小化办法,其中既包括一级和二级变式。两套算法都有理论保证。特别是,对于交替最小化,我们建立了全球趋同和最坏情况复杂界限。此外,我们使用Kurdyka-Lojasiewicz的属性,我们表明,交替最小化将汇集到一个独特的限制点。我们为回收子空间的结合和集聚集在进入取样和密集高斯抽样中提供了广泛的数字结果。我们的方法与现有方法具有竞争力,特别是利用里曼二级方法在回收中达到很高的精度。