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Partial singular value decomposition via Lanczos method for dual quaternion matrices and applications

This paper proposes a numerically stable and efficient partial singular value decomposition method for dual quaternion matrices using the Lanczos algorithm with thick-restart and one-sided reorthogonalization, demonstrating its superior accuracy and computational performance in applications such as color face recognition and video compression.

Original authors: Yizhe Zhang, Ying Li

Published 2026-08-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yizhe Zhang, Ying Li

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of digital information, images and videos are not just pictures; they are vast grids of numbers. When we look at a color photograph, a computer sees three separate layers of intensity—red, green, and blue—stacked together. To understand these images, to compress them for storage, or to recognize a face within them, mathematicians often rely on a powerful tool called singular value decomposition. Think of this process as a way to sift through a massive pile of data to find the most important threads that hold the picture together, while ignoring the noise. For decades, this tool has worked well for simple black-and-white images or for color images treated as separate, disconnected lists of numbers. However, treating the colors as separate strips ignores the natural way they work together as a single, unified whole. To solve this, researchers have turned to a more complex mathematical system known as dual quaternions, which allows them to handle the standard parts of a number and its tiny, infinitesimal variations simultaneously, keeping the relationships between colors intact.

The challenge, however, is that working with these complex, unified numbers is incredibly slow. Calculating the full set of important patterns for a large image or a long video using these advanced methods can take so much computing power that it becomes impractical. Often, we do not need every single detail; we only need the most dominant patterns to get the job done. A team of researchers at Liaocheng University has developed a new method to speed this process up significantly. They created a technique that finds only the most essential patterns in these complex color data sets, rather than calculating the entire, overwhelming picture. Their approach, which they call a partial singular value decomposition, uses a specific mathematical strategy to project the massive data onto a much smaller, manageable space. By doing this, they can extract the key features needed for tasks like identifying a person's face or shrinking a video file, without the heavy computational cost that usually comes with such high-level math.

The researchers tested their new method on two very different real-world problems: recognizing faces in color photographs and compressing color video footage. For the face recognition task, they treated each color image as a single, unified block of data rather than splitting it into separate red, green, and blue channels. This allowed them to preserve the spatial structure of the face, keeping the relationships between the eyes, nose, and mouth intact in a way that traditional methods often lose. When they applied their new algorithm to a database of fifty different individuals, the results were striking. The new method not only recognized faces more accurately than previous approaches but also did so much faster. In their tests, the algorithm consistently outperformed older methods, reaching high levels of accuracy with less computing time, proving that keeping the color information unified leads to better and quicker results.

For video compression, the team adapted their method to handle a slightly different type of complex number, which is a special case of the system they used for faces. They took a short video clip consisting of six frames and used their algorithm to find the most important patterns that define the movement and color of the scene. The goal was to see if they could shrink the file size while keeping the picture looking clear. The results showed that their new technique was significantly faster than existing methods. While the mathematical scores for image quality were slightly lower than those of the older method, the visual difference was negligible. To the human eye, the compressed video looked just as good as the original, yet the computer finished the job in a fraction of the time. This suggests that for applications where speed is critical, such as streaming video or processing large archives, this new approach offers a practical and efficient solution.

The success of this work lies in how it handles the math behind the scenes. Instead of trying to solve the entire problem at once, the researchers broke it down into smaller steps, focusing only on the parts of the data that mattered most. They used a strategy that allowed them to restart and refine their calculations without losing the progress they had already made, ensuring that the process remained stable and accurate. By combining these techniques, they managed to tame the complexity of dual quaternion matrices, turning a computationally expensive task into something that can be done quickly and reliably. The findings confirm that it is possible to work with these advanced, unified mathematical models without sacrificing speed, opening the door for more efficient processing of color images and videos in the future.

Technical Summary: Partial Singular Value Decomposition via Lanczos Method for Dual Quaternion Matrices and Applications

1. Problem Statement

Singular Value Decomposition (SVD) is a fundamental tool for low-rank approximation, yet computing the full SVD for large-scale matrices is often computationally prohibitive. In many practical applications, only a few dominant singular values are required to capture the essential information of the data. While existing algorithms for Dual Quaternion (DQ) matrices typically focus on computing the full decomposition, there is a lack of efficient methods to compute partial SVDs for large-scale DQ data. This limitation hinders the application of DQ matrices in fields requiring high-dimensional data processing, such as color image and video analysis, where computational efficiency is critical.

2. Methodology

The authors propose lansvdDQ, a partial SVD algorithm for dual quaternion matrices based on the Lanczos bidiagonalization method. The approach consists of three main components:

2.1 Lanczos Bidiagonalization for Dual Quaternion Matrices

The method extends the Golub-Kahan-Lanczos process to the dual quaternion domain. Given a dual quaternion matrix $A \in \mathbb{DQ}^{m \times n}$ , the algorithm projects $A$ onto a Krylov subspace to generate a much smaller bidiagonal matrix $B_k$ .

Process: Starting with a unit vector, the algorithm iteratively computes left and right Lanczos vectors ( $u_i, v_i$ ) and dual numbers ( $\alpha_i, \beta_i$ ) to form $B_k = U_k^* A V_k$ .
Stability: To address the loss of orthogonality due to finite precision arithmetic, a one-sided reorthogonalization strategy is employed. This strategy enforces orthogonality specifically on the right Lanczos vectors ( $v_i$ ), which is sufficient to maintain the stability of the coupled left vectors through the bidiagonal structure.

2.2 Low-Rank Approximation via Krylov Subspace

Once the bidiagonal matrix $B_k$ is constructed, its SVD is computed to obtain Ritz singular triplets $(\sigma_i, \tilde{u}_i, \tilde{v}_i)$ . These serve as approximations for the dominant singular triplets of the original matrix $A$ . The low-rank approximation $A_k$ is reconstructed as:
$A_k = \tilde{U}_k \Sigma_k \tilde{V}_k^*$
The quality of the approximation is evaluated using residual norms derived from the bidiagonalization process.

2.3 Thick-Restart Technique

To improve convergence and avoid restarting the entire process from scratch, a thick-restart strategy is incorporated.

Mechanism: When $l$ Ritz triplets have converged (out of $k$ iterations), the algorithm retains these converged vectors in the basis. It then appends the last Lanczos vector (carrying remaining spectral information) to construct a new basis.
Execution: The Lanczos process continues from this enriched basis, allowing the number of converged triplets to increase to the target $s$ without discarding useful spectral information from previous iterations.

3. Key Contributions

Algorithm Development: The paper extends the Lanczos-based partial SVD approach to the dual quaternion setting, introducing the lansvdDQ algorithm.
Numerical Stability and Efficiency: By integrating one-sided reorthogonalization and thick-restart techniques, the method ensures numerical stability and accelerates convergence for large-scale problems.
Application Frameworks: The authors adapt the method to two distinct practical scenarios:
1. Color Face Recognition: Utilizing a 2D approach within the dual quaternion framework to preserve spatial structure and encode RGB/HSV information simultaneously.
2. Color Video Compression: Adapting the method to the dual complex setting (a special case of dual quaternions) to reduce algebraic complexity while maintaining visual quality.

4. Experimental Results

The proposed lansvdDQ was compared against the existing partial SVD method for quaternion matrices (lansvdQ) [12] using standard benchmarks.

4.1 Face Recognition

Dataset: Georgia Tech face database (50 subjects, 15 images each).
Performance:
- Accuracy: lansvdDQ achieved higher recognition accuracy than lansvdQ across all tested dimensions ( $d$ ).
- Convergence: The recognition rate of lansvdDQ converged more rapidly to a stable value.
- Efficiency: Due to the 2D approach operating on $256 \times 256$ covariance matrices rather than high-dimensional vectors, lansvdDQ demonstrated significantly lower CPU times compared to lansvdQ.

4.2 Video Compression

Dataset: A custom "fish.mp4" video sequence (6 frames, $480 \times 760$ pixels).
Method: The video was modeled as a dual complex matrix ( $A \in \mathbb{DC}^{2880 \times 760}$ ).
Performance:
- Speed: The dual complex variant (lansvdDC) was consistently faster than lansvdQ across all retained singular triplet counts ( $d = 20$ to $40$).
- Quality: While quantitative metrics (PSNR, SSIM) for lansvdDC were slightly lower than lansvdQ, the visual reconstruction quality was comparable. With $d=30$ , reconstructed frames were visually indistinguishable from the original.
- Conclusion: The method offers a practical trade-off, prioritizing computational efficiency without significant loss in visual fidelity.

4.3 Numerical Validation

Experiments on randomly generated dual quaternion matrices showed that the relative $FR$ -norm error decreased as the number of retained singular triplets increased, confirming the algorithm's ability to effectively extract dominant spectral information.

5. Significance and Claims

The paper claims that lansvdDQ provides an efficient and practical solution for the low-rank approximation of large-scale dual quaternion data. By avoiding the computational expense of full SVD and leveraging the structural advantages of dual quaternions (and their special case, dual complex numbers), the method enables:

Higher Accuracy: Improved performance in color face recognition by preserving spatial structure and color correlations.
Lower Computational Cost: Reduced processing time for both face recognition and video compression tasks compared to existing quaternion-based approaches.
Scalability: The ability to handle large-scale data where full decomposition is infeasible, making it suitable for real-world applications in image and video processing.

The authors conclude that the integration of Lanczos bidiagonalization with thick-restart and reorthogonalization strategies successfully addresses the challenges of computing partial SVDs in the dual quaternion domain.