Concentration bounds for modewise tensor sketching

Multiplication by a wide random matrix is a well-established tool in the data dimension reduction, featured in the Johnson-Lindenstrauss lemma and followed by many related results. Randomized algorithms on multimodal tensorial data often compress the data mode-by-mode, aiming to produce a smaller tensor rather than unstructured measurements. Mathematically, it results in Kronecker-structured random sketching matrices, which suffer from heavy-tailed behavior and dependencies in the resulting sketching matrix even when the components were independent Gaussian matrices. In particular, their success for tensor data compression — that depends on the moments of the sketched tensor -- can be poor on very simple rank-one tensorial objects. But is this actually one of the worst cases?