Quaternion tensor singular value decomposition using a flexible transform-based approach

Abstract: A flexible transform-based tensor product named $\star_{{\rm{QT}}}$-product for $L$th-order ($L\geq 3$) quaternion tensors is proposed. Based on the $\star_{{\rm{QT}}}$-product, we define the corresponding singular value decomposition named TQt-SVD and the rank named TQt-rank of the $L$th-order ($L\geq 3$) quaternion tensor. Furthermore, with orthogonal quaternion transformations, the TQt-SVD can provide the best TQt-rank-$s$ approximation of any $L$th-order ($L\geq 3$) quaternion tensor. In the experiments, we have verified the effectiveness of the proposed TQt-SVD in the application of the best TQt-rank-$s$ approximation for color videos represented by third-order quaternion tensors.

• The paper introduces the TQt-SVD method, offering a flexible transform-based approach for quaternion tensor decomposition applied to higher-order data.
• It defines a new quaternion tensor product, the star_QT-product, which generalizes traditional tensor multiplication to effectively approximate data.
• Experimental results demonstrate enhanced PSNR in color video processing through adaptive transformations and optimal rank-s approximations.

Quaternion Tensor Singular Value Decomposition

Introduction

The paper "Quaternion tensor singular value decomposition using a flexible transform-based approach" presents a novel framework for singular value decomposition (SVD) applied to quaternion tensors. Unlike traditional real and complex matrices, quaternion matrices have components extending in four dimensions, allowing them to capture richer structural information, especially relevant in color video processing where the channels are correlated. The paper addresses limitations of prior quaternion tensor SVD methods by introducing a transform-based product adaptable to tensors of arbitrary $L$-th order ($L \geq 3$), using flexible quaternion transforms.

Quaternion Tensor Product and Transform

The central innovation of this work is the $\star_{\text{QT}}$-product, generalizing the concept of tensor products to utilize any invertible quaternion transformation. This product leverages quaternion tensor properties to define TQt-SVD (Transform-based Quaternion Tensor-Singular Value Decomposition), providing best rank-$s$ approximations for higher-order tensors without restricting transformations to fixed domains like the Discrete Fourier Transform (DFT).

The novel product, $\star_{\text{QT}}$-product, multiplies quaternion tensors through facewise and transform-based operations, facilitating flexible decomposition strategies and enabling applications across diverse data types beyond third-order tensors.

Theoretical Contributions

The paper establishes several foundational definitions and propositions for quaternion tensor operations:

• Quaternion facewise product: A basic operation defined for element-wise multiplication across frontal slices of tensors.
• Conjugate transpose, identity quaternion tensor, unitary quaternion tensor: These definitions extend classical matrix concepts to quaternion tensors under the $\star_{\text{QT}}$-product framework.
• TQt-SVD: It provides a decomposition into unitary quaternion tensors, facilitating dimensionality-reduction in high-order data spaces.

The authors propose the TQt-rank, determined by the significant singular values of the decomposition, and prove the optimality of TQt-rank-$s$ approximations in preserving tensor structure.

Practical Implications

The effectiveness of TQt-SVD is validated through experiments on color video processing. By representing videos as third-order quaternion tensors where RGB channels map to quaternion parts, TQt-SVD enables adaptive decomposition leading to efficient storage and representation. Experimental results demonstrate superior Peak Signal-to-Noise Ratio (PSNR) performance compared to traditional Qt-SVD methods, particularly when flexible data-driven transformations are employed.

Moreover, the ability to choose transformations on-the-fly offers a significant advantage in tailoring decompositions based on specific characteristics of the data, enhancing robustness and accuracy in applications like video compression and enhancement.

Conclusion

The study introduces a flexible, transform-based method for quaternion tensor SVD capable of handling higher-order structures. The flexibility in choosing transformations presents a potent tool for adapting SVDs to specific data properties, ultimately promising improved performance in practical applications. Future research may explore further refinements in selecting optimal transformations for diverse datasets and extending similar frameworks to other tensor decomposition realms, potentially impacting computer vision, signal processing, and multi-channel data analytics.