Sketching Low-Rank Matrices With a Shared Column Space by Convex Programming
Sketching Low-Rank Matrices With a Shared Column Space by Convex Programming
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DOI：
10.1109/jsait.2023.3283973
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发表时间：
2022-10
期刊：
影响因子：
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通讯作者：
R. S. Srinivasa;Seonho Kim;Kiryung Lee
中科院分区：
文献类型：
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作者：
R. S. Srinivasa;Seonho Kim;Kiryung Lee
在许多实际应用中，包括遥感、多任务学习和多光谱成像，数据被描述为共享公共列空间的一组矩阵。我们考虑了这类矩阵的噪声线性测量的联合估计。我们研究由一对矩阵范数正则化的凸估计量。该测量模型对应于逐块感测，并且仅当总能量在块上均匀分布时才可能进行重建。第一个范数，即最大块-Frobenius范数，支持这样的解。这种情况类似于矩阵补全或列感测中的低尖峰概念。第二个范数是一对适当的Banach空间上的张量范数，它与第一个范数一起在解中诱导低秩性。我们证明了当共享列空间的矩阵的数目和共享列空间的环境维度相对于每个矩阵中的列数较大时，联合估计相对于每个矩阵的单独恢复提供了显著的增益。将凸估计转化为半定规划，给出了一种有效的ADMM算法。利用蒙特卡罗模拟说明了凸估计的经验行为，并将恢复性能与文献中已有的方法进行了比较。
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.