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.

翻译：我们针对从线性观测中恢复多个异质低维结构数据的问题，提出了一种新算法。聚焦于同时满足行稀疏性和低秩性的数据矩阵，我们设计并分析了一种能够充分利用这两种结构的迭代重加权最小二乘（IRLS）算法。具体而言，该算法优化了行稀疏性和秩的非凸替代函数的组合，并将两者的平衡机制内置到算法中。我们证明，在最小样本复杂度（常数和对数因子内）条件下，该算法的迭代结果以局部二次收敛速度收敛至同步结构化数据矩阵，而已知使用凸替代函数的组合无法实现这一结果。实验表明，IRLS方法在经验收敛性方面表现优异，能够从比现有方法更少的测量数据中识别出同时具有行稀疏性和低秩性的矩阵。