Modewise Operators, the Tensor Restricted Isometry Property, and Low-Rank Tensor Recovery
Modewise Operators, the Tensor Restricted Isometry Property, and Low-Rank Tensor Recovery
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DOI：
10.1016/j.acha.2023.04.007
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发表时间：
2021-09
期刊：
影响因子：
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通讯作者：
M. Iwen;D. Needell;Michael Perlmutter;E. Rebrova
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文献类型：
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作者：
M. Iwen;D. Needell;Michael Perlmutter;E. Rebrova
众所周知，在关于测量的各种模型假设下，从少量的线性测量中恢复稀疏向量和低秩阵是可能的。对测量矩阵的关键要求通常是受限等距性质，即作用于待恢复的子空间时的近似正交性。最广泛使用的随机矩阵测量模型有(A)独立的亚高斯模型和(B)基于随机傅立叶的模型，允许测量的有效计算。对于现在普遍存在的张量数据，由于要构造和存储巨大的测量矩阵，直接将已知的恢复算法应用于矢量化或矩阵化张量需要耗费大量内存。在这篇文章中，我们提出了基于亚高斯和随机傅立叶测量的模态化测量方案。这些模式操作符分别作用于张量模式的对或其他小的子集。它们比在矢量化张量上工作的测量所需的内存要少得多，可以证明满足张量受限等距性质，并且实验上可以从更少的测量中恢复张量数据，并且不需要不切实际的存储。
Recovery of sparse vectors and low-rank matrices from a small number of linear measurements is well-known to be possible under various model assumptions on the measurements. The key requirement on the measurement matrices is typically the restricted isometry property, that is, approximate orthonormality when acting on the subspace to be recovered. Among the most widely used random matrix measurement models are (a) independent subgaussian models and (b) randomized Fourier-based models, allowing for the efficient computation of the measurements.For the now ubiquitous tensor data, direct application of the known recovery algorithms to the vectorized or matricized tensor is memory-heavy because of the huge measurement matrices to be constructed and stored. In this paper, we propose modewise measurement schemes based on subgaussian and randomized Fourier measurements. These modewise operators act on the pairs or other small subsets of the tensor modes separately. They require significantly less memory than the measurements working on the vectorized tensor, provably satisfy the tensor restricted isometry property and experimentally can recover the tensor data from fewer measurements and do not require impractical storage.