The wireless communication channel usually spans several physical 'dimensions' such as space, time, frequency, polarization, etc. Numerous works have successfully used tensor decompositions as a mathematical formalism to describe the algebraic structure of the wireless channel.

Tensor decompositions have recently been exploited to model and estimate mmWave channels, assuming hybrid analog-digital architectures. Leveraging tridimensional sparse representations of the large-scale channel matrices, tensor compressive sensing (tensor-CS) methods can effectively estimate the channel parameters by compressively sampling them in space (radio-frequency chains), time (symbol periods) and frequency (pilot subcarriers).

Alternatively, the wireless channel can be algebraically represented with tensor train representations, where the higher-order tensor channel is decomposed into third-order smaller tensors (each one being a 'wagon' of the tensor train). The tensor train approach significantly reduces the number of channel coefficients estimated by exploiting the low-rank channel structure if available. This is particularly useful when dealing with large-scale and structured MIMO channels in higher frequency bands.