We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogeneous low-dimensional structures from linear observations. Focusing on data matrices that are simultaneously row-sparse and low-rank, we propose and analyze an iteratively reweighted least squares (IRLS) algorithm that is able to leverage both structures. In particular, it optimizes a combination of non-convex surrogates for row-sparsity and rank, a balancing of which is built into the algorithm. We prove locally quadratic convergence of the iterates to a simultaneously structured data matrix in a regime of minimal sample complexity (up to constants and a logarithmic factor), which is known to be impossible for a combination of convex surrogates. In experiments, we show that the IRLS method exhibits favorable empirical convergence, identifying simultaneously row-sparse and low-rank matrices from fewer measurements than state-of-the-art methods. Code is available at https://github.com/ckuemmerle/simirls.

翻译：我们提出一种新算法，用于从线性观测中恢复符合多种异质低维结构的数据。针对同时具有行稀疏性和低秩性的数据矩阵，我们提出并分析了一种能够同时利用这两种结构的迭代重加权最小二乘（IRLS）算法。该算法优化了行稀疏性和秩的非凸替代函数的组合，其平衡策略已内置于算法中。我们证明了在最小样本复杂度（常数和对数因子范围内）条件下，迭代过程以局部二次收敛速度收敛至同时结构化数据矩阵——已知仅使用凸替代函数的组合无法实现该收敛速度。实验表明，IRLS方法具有优越的经验收敛性能，能以比现有方法更少的测量次数识别同时具有行稀疏性和低秩性的矩阵。代码已发布于https://github.com/ckuemmerle/simirls。