  Tensors of order three or higher have found applications in diverse fields,
including image and signal processing, data mining, biomedical engineering and
link analysis, to name a few. In many applications that involve for example
time series or other ordered data, the corresponding tensor has a
distinguishing orientation that exhibits a low tubal structure. This has
motivated the introduction of the tubal rank and the corresponding tubal
singular value decomposition in the literature. In this work, we develop
randomized algorithms for many common tensor operations, including tensor
low-rank approximation and decomposition, together with tensor multiplication.
The proposed tubal focused algorithms employ a small number of lateral and/or
horizontal slices of the underlying 3-rd order tensor, that come with {\em
relative error guarantees} for the quality of the obtained solutions. The
performance of the proposed algorithms is illustrated on diverse imaging
applications, including mass spectrometry data and image and video recovery
from incomplete and noisy data. The results show both good computational
speed-up vis-a-vis conventional completion algorithms and good accuracy.
