  In CS literature, the efforts can be divided into two groups: finding a
measurement matrix that preserves the compressed information at the maximum
level, and finding a reconstruction algorithm for the compressed information.
In the traditional CS setup, the measurement matrices are selected as random
matrices, and optimization-based iterative solutions are used to recover the
signals. However, when we handle large signals, using random matrices become
cumbersome especially when it comes to iterative optimization-based solutions.
Even though recent deep learning-based solutions boost the reconstruction
accuracy performance while speeding up the recovery, still jointly learning the
whole measurement matrix is a difficult process. In this work, we introduce a
separable multi-linear learning of the CS matrix by representing it as the
summation of arbitrary number of tensors. For a special case where the CS
operation is set as a single tensor multiplication, the model is reduced to the
learning-based separable CS; while a dense CS matrix can be approximated and
learned as the summation of multiple tensors. Both cases can be used in CS of
two or multi-dimensional signals e.g., images, multi-spectral images, videos,
etc. Structural CS matrices can also be easily approximated and learned in our
multi-linear separable learning setup with structural tensor sum
representation. Hence, our learnable generalized tensor summation CS operation
encapsulates most CS setups including separable CS, non-separable CS
(traditional vector-matrix multiplication), structural CS, and CS of the
multi-dimensional signals. For both gray-scale and RGB images, the proposed
scheme surpasses most state-of-the-art solutions, especially in lower
measurement rates. Although the performance gain remains limited from tensor to
the sum of tensor representation for gray-scale images, it becomes significant
in the RGB case.
