Fourth-order Tensors with Multidimensional Discrete Transforms

TL;DR

This paper introduces a novel multilinear tensor space for fourth-order tensors supporting algorithms like SVD and QR, generalizing matrix concepts to higher dimensions with applications in video compression and face recognition.

Contribution

The paper develops a new tensor space based on multidimensional discrete transforms, defining tensor operations and decompositions such as $\\mathcal{L}$-SVD and QR, extending matrix methods to fourth-order tensors.

Findings

The $\mathcal{L}$-SVD improves video compression by 3-10dB in RSE and reduces computation time.

The tensor $\mathcal{L}$-QR algorithm avoids cancellation issues and enhances stability.

Recognition rate in face recognition increases by 10-20%.

Abstract

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 $10\% \sim 20\%$.