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Recovery of sparse vectors and low-rank matrices from a small number oflinear measurements is well-known to be possible under various modelassumptions on the measurements. The key requirement on the measurementmatrices is typically the restricted isometry property, that is, approximateorthonormality when acting on the subspace to be recovered. Among the mostwidely used random matrix measurement models are (a) independent sub-gaussianmodels and (b) randomized Fourier-based models, allowing for the efficientcomputation of the measurements.  For the now ubiquitous tensor data, direct application of the known recoveryalgorithms to the vectorized or matricized tensor is awkward and memory-heavybecause of the huge measurement matrices to be constructed and stored. In thispaper, we propose modewise measurement schemes based on sub-gaussian andrandomized Fourier measurements. These modewise operators act on the pairs orother small subsets of the tensor modes separately. They require significantlyless memory than the measurements working on the vectorized tensor, provablysatisfy the tensor restricted isometry property and experimentally can recoverthe tensor data from fewer measurements and do not require impractical storage.

Modewise Operators, the Tensor Restricted Isometry Property, and Low-Rank Tensor Recovery