In many practical applications including remote sensing, multi-task learning, and multi-spectrum imaging, data are described as a set of matrices sharing a common column space. We consider the joint estimation of such matrices from their noisy linear measurements. We study a convex estimator regularized by a pair of matrix norms. The measurement model corresponds to block-wise sensing and the reconstruction is possible only when the total energy is well distributed over blocks. The first norm, which is the maximum-block-Frobenius norm, favors such a solution. This condition is analogous to the notion of low-spikiness in matrix completion or column-wise sensing. The second norm, which is a tensor norm on a pair of suitable Banach spaces, induces low-rankness in the solution together with the first norm. We demonstrate that the joint estimation provides a significant gain over the individual recovery of each matrix when the number of matrices sharing a column space and the ambient dimension of the shared column space are large relative to the number of columns in each matrix. The convex estimator is cast as a semidefinite program and an efficient ADMM algorithm is derived. The empirical behavior of the convex estimator is illustrated using Monte Carlo simulations and recovery performance is compared to existing methods in the literature.

翻译：在许多实际应用中，包括遥感、多任务学习和多光谱成像，数据被描述为一组共享共同列空间的矩阵。我们考虑从这些矩阵的带噪线性测量中联合估计它们。我们研究了一种由一对矩阵范数正则化的凸估计器。测量模型对应于分块采样，只有当总能量在块间良好分布时，重构才可能实现。第一个范数是最大块-弗罗贝尼乌斯范数，它有利于这种解。该条件类似于矩阵补全或列采样中的低尖峰性概念。第二个范数是一对适当巴拿赫空间上的张量范数，与第一个范数共同诱导解的低秩性。我们证明，当共享列空间的矩阵数量及共享列空间的维度相对于各矩阵的列数较大时，联合估计相较于单独恢复每个矩阵能提供显著增益。该凸估计器被表述为半定规划，并推导出高效的ADMM算法。通过蒙特卡罗仿真说明了凸估计器的经验性能，并将其恢复性能与文献中的现有方法进行了比较。