Speaker
Description
Singular value decomposition (SVD) is a fundamental tool in data analysis and machine learning. Starting from the Stewart’s QLP decomposition [1], we propose an innovative Deep-QLP decomposition algorithm for efficiently computing an approximate Singular Value Decomposition (SVD) based on the preliminary work in [2]. Given a specified tolerance $\tau$, the algorithm automatically computes a positive integer $f$ and a factorization $\mathcal{U}_f \mathcal{L}_f^D \mathcal{V}_f^T$, with $\mathcal{L}_f^{D}$ diagonal matrix, $\mathcal{U}_f,\mathcal{V}_f$ matrices of rank $f$ with orthonormal columns such that $
\|A-\mathcal{U}_{f}\mathcal{L}^{D}_{f}\mathcal{V}_{f}^{T}\|_2 \leq 3\tau \|A\|_2.$
The Deep-QLP algorithm stands out for its ability to return an approximation of the largest singular values, based on a fixed tolerance, to achieve significant dimensionality reduction while simultaneously preserving essential information in the data. In addition, it can also be used to return an approximation of the smallest singular values that can be used in some applications.
The algorithm has been successfully integrated with the randomized SVD [3], making
the Deep-QLP algorithm particularly effective for sparse matrices, which are prevalent
in numerous applications such as text mining.
Several numerical experiments have been conducted, demonstrating the effectiveness of the proposed method.
References
- Gilbert W. Stewart. The QLP approximation to the singular value decomposition. SIAM J. Sci. Comput., 20:1336–1348, 1999.
- Antonella Falini and Francesca Mazzia. Approximated iterative QLP for change detection in hyperspectral images. AIP Conference Proceedings, 3094(1):370003, 06 2024.
- Nathan Halko, Per-Gunnar Martinsson, and Joel A Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53(2):217–288, 2011.
