Sensing Tensors With Gaussian Filters

Sparse recovery from linear Gaussian measurements has been the subject of much investigation since the breakthrough papers Candès et al. and Donoho on compressed sensing. Application to sparse vectors and sparse matrices via least squares penalized with sparsity promoting norms is now well understood thanks to tools, such as Gaussian mean width, statistical dimension, and the notion of descent cones. Extension of these ideas to low rank tensor recovery is starting to enjoy considerable interest due to its many potential applications to independent component analysis, hidden Markov models, Gaussian mixture models, and hyperspectral image analysis, to name a few. In this paper, we demonstrate that the recent approach of Vershynin provides useful error bounds in the tensor setting with the nuclear norm or the Romera-Paredes-Pontil penalization.