  The big data era is swamping areas including data analysis, machine/deep
learning, signal processing, statistics, scientific computing, and cloud
computing. The multidimensional feature and huge volume of big data put urgent
requirements to the development of multilinear modeling tools and efficient
algorithms. In this paper, we build a novel multilinear tensor space that
supports useful algorithms such as SVD and QR, while generalizing the matrix
space to fourth-order tensors was believed to be challenging. Specifically,
given any multidimensional discrete transform, we show that fourth-order
tensors are bilinear operators on a space of matrices. First, we take a
transform-based approach to construct a new tensor space by defining a new
multiplication operation and tensor products, and accordingly the analogous
concepts: identity, inverse, transpose, linear combinations, and orthogonality.
Secondly, we define the $\mathcal{L}$-SVD for fourth-order tensors and present
an efficient algorithm, where the tensor case requires a stronger condition for
unique decomposition than the matrix case. Thirdly, we define the tensor
$\mathcal{L}$-QR decomposition and propose a Householder QR algorithm to avoid
the catastrophic cancellation problem associated with the conventional
Gram-Schmidt process. Finally, we validate our schemes on video compression and
one-shot face recognition. For video compression, compared with the existing
tSVD, the proposed $\mathcal{L}$-SVD achieves $3\sim 10$dB gains in RSE, while
the running time is reduced by about $50\%$ and $87.5\%$, respectively. For
one-shot face recognition, the recognition rate is increased by about $10\%
\sim 20\%$.
