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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 sub-Gaussian models and (b) randomized Fourier-based models, allowing for the efficient computation of the measurements.

The tensor structure of the data leads to challenges both in defining low dimensional structure as well as recovery techniques (direct application of the known recovery algorithms to the vectorized or matricized tensor is awkward and memory-heavy). I will discuss modewise (structure-preserving) measurement schemes based on sub-Gaussian and randomized Fourier measurements. They are significantly smaller than the measurements working on the vectorized tensors, provably satisfy the restricted isometry property, and experimentally can recover the tensor data from fewer measurements.