Impact of Singular Value Decomposition: A review study

 Corresponding author: Atul Thomas Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution Liscense 4.0. Impact of Singular Value Decomposition: A review study Atul Thomas * and Mithila Chandra Department of Computer Science, Student, Osmania University, Hyderabad, India. World Journal of Advanced Research and Reviews, 2025, 28(01), 1198-1202 Publication history: Received on 09 September 2025; revised on 14 October 2025; accepted on 17 October 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.28.1.3552 Abstract Singular Value Decomposition (SVD) is a fundamental matrix factorization technique that provides deep insight into the structure of linear systems. It decomposes a given matrix into orthogonal and diagonal components, enabling the identification of key features such as rank, range, and noise characteristics. This paper discusses both the computational methods for obtaining the SVD and its wide-ranging applications across science and engineering. Keywords: Singular Value Decomposition; Eigen values; Rank; Transform 1. Introduction Singular Value Decomposition (SVD) is a fundamental mathematical technique in linear algebra that decomposes a matrix into three component matrices representing its intrinsic geometric structure. In imaging applications, this decomposition separates signal components based on their relative strength or significance, allowing selective suppression of noise and enhancement of meaningful features. By isolating dominant singular values associated with true signal content and filtering out smaller singular values corresponding to noise, SVD provides an effective framework for improving image quality. Its versatility and interpretability have made it a valuable tool in various domains, including signal processing, pattern recognition, and medical image reconstruction[1, 2]. In the context of medical imaging, particularly in angiography, SVD has emerged as a promising method for improving the accuracy of quantitative measurements. The technique helps stabilize image-derived parameters by mitigating the effects of acquisition noise, injection variability, and motion artifacts—common challenges in vascular imaging[3, 4]. Unlike conventional filtering methods that may blur fine vascular details, SVD preserves spatial and temporal integrity while enhancing the signal-to-noise ratio (SNR). As a result, it enables more consistent extraction of hemodynamic and morphological features critical for clinical assessment. Given its data-driven nature and adaptability, SVD offers a robust foundation for developing standardized and reproducible image-processing pipelines in both research and clinical settings[5-7]. This paper takes the SVD formulation from Mondal et al.’s [8] paper on deconvolutional methods using SVD and brings out the analysis on the mathematical formulation of the method. SVD helps to reduce noise and variability, improve the signal-to-noise ratio (SNR), and enhance the reliability of vascular measurements in 2D angiography. The findings of this review help understanding the concept of SVD which can be helpful for both clinical practice and research[9, 10]. World Journal of Advanced Research and Reviews, 2025, 28(01), 1198-1202 1199 2. Materials and Methods 2.1. Definition Singular Value Decomposition (SVD) is a matrix factorization technique that expresses any real (or complex) matrix as the product of three special matrices: A = U ∑ VT where: A is an m x n matrix U is an m x m orthogonal matrix (UTU = Im) V is an n x n orthogonal matrix (VTV = In) ∑ is a m x n diagonal matrix whose diagonal entries are non-negative and called singular values. Mathematically, Where σ1 ≥ σ2 ≥ σ3 … ≥ σp ≥ 0, and p = min(m,n). 2.2. Mathematical derivation To find the SVD, start with the symmetric matrices ATA and AAT : • Compute eigen values and eigen vectors of ATA: ATA vi = λi vi Where λi ≥ 0 are eigen values and vi are eigen vectors of ATA Define σi = √λi these are the singular values • Similarly, A AT ui = λi ui gives singular vectors ui Hence, A vi = σi ui , AT ui = vi σi Assembling ui into u, all vi into V, and all σi into ∑, we get: A = U ∑ VT 2.3. Geometric Interpretation SVD can be interpreted geometrically as: A : Rn Rm World Journal of Advanced Research and Reviews, 2025, 28(01), 1198-1202 1200 transforms the unit sphere in Rn into an ellipsoid in Rm. V rotates (or reorients) the coordinate system in input space. Σ stretches or compresses along orthogonal directions by factors σi U rotates again to align the output in the target space. 2.4. Rank and Approximation If the rank of A is r, then only the first r singular values are nonzero. SVD provides the best low-rank approximation to A in terms of the Frobenius norm or 2-norm: This Ar minimizes Among all rank-r matrices. This property is the foundation of Principal Component Analysis (PCA) and image compression. 2.5. Numerical stability and Applications Because SVD works even for non-square and ill-conditioned matrices, it’s numerically stable and widely used in: • Least-squares solutions: For Ax = b, • Noise reduction and filtering: remove components corresponding to small singular values. • Data compression and PCA: retain only top singular values. • Image denoising, medical imaging (CT/MRI reconstruction), and signal processing[11]. Example: For example: if Then World Journal of Advanced Research and Reviews, 2025, 28(01), 1198-1202 1201 Whose eigen values are λ1 = 16, λ2 = 4. Hence, σ1 = 4, σ2 = 2. The SVD of A can be written as: Showing how A stretches space by 4x and 2x along orthogonal directions. Limitations Some of the limitations of SVD are as follows: • Computing the SVD via the eigen value decomposition of ATA is numerically unstable because it squares the condition number of A [12]. • SVD is computationally expensive, especially for large matrices. • When dealing with ill-conditioned matrices, small perturbations in data can lead to large changes in the smallest singular values. • Full SVD requires storage of three large matrices U, Σ, VT. • For high-dimensional problems, this results in substantial memory usage. • Efficient partial or truncated SVD algorithms exist, but their implementation is complex and sensitive to numerical precision. The accuracy of small singular values is limited by machine arithmetic. For example, a matrix with a smallest singular value on the order of 10-12 may appear as 10-8 or vanish entirely, depending on floating-point representation, which can mislead interpretation of rank or stability. While SVD is a powerful and elegant mathematical tool for analyzing matrix structure, its practical limitations stem from computational cost, numerical sensitivity, and finite precision effects on modern computing machines 3. Conclusion Singular Value Decomposition breaks a matrix into its most fundamental components — rotations and scalings — revealing how it transforms space. Mathematically elegant and computationally stable, SVD underpins modern applications from AI-driven imaging to recommendation systems and genome analysis. Compliance with ethical standards Acknowledgments The Authors would like to thank the heads of the institution, teachers, staff and students for their co-operation and support. Disclosure of conflict of interest No conflict of interest to be disclosed. Statement of ethical approval The study was approved by the Institutional Ethics Committee World Journal of Advanced Research and Reviews, 2025, 28(01), 1198-1202 1202 Statement of informed consent Informed consent was obtained from all participants included in the study References [1] Weiss S, Proudler IK, Barbarino G, Pestana J, McWhirter JG: On Properties and Structure of the Analytic Singular Value Decomposition. IEEE Transactions on Signal Processing 2024, 72:2260-2275. [2] Jolliffe IT, Cadima J: Principal component analysis: a review and recent developments. Philos Trans A Math Phys Eng Sci 2016, 374(2065):20150202. [3] Mondal P, Shields A, Bhurwani MMS, Williams KA, Veeturi SS, Nagesh SVS, Siddiqui AH, Ionita CN: Analysis of quantitative angiography in intracranial aneurysm using projection foreshortening correction and injection bias removal. Medical Physics 2025, 52(7):e17965. [4] Klema V, Laub A: The singular value decomposition: Its computation and some applications. IEEE Transactions on Automatic Control 1980, 25(2):164-176. [5] Lin C, Li L, Luo W, Wang KC, Guo J: Transfer learning based traffic sign recognition using inception-v3 model. Periodica Polytechnica Transportation Engineering 2019, 47(3):242-250. [6] Steinberg MH: Predicting clinical severity in sickle cell anaemia. British journal of haematology 2005, 129(4):465-481. [7] Tio AE: Face shape classification using inception v3. arXiv preprint arXiv:191107916 2019. [8] Mondal P, Setlur Nagesh SV, Sommers-Thaler S, Shields A, Shiraz Bhurwani MM, Williams KA, Baig A, Snyder K, Siddiqui AH, Levy E, Ionita CN: Effect of singular value decomposition on removing injection variability in 2D quantitative angiography: An in silico and in vitro phantoms study. Medical Physics 2024, 51(11):8192-8212. [9] Peri E, Xu L, Ciccarelli C, Vandenbussche NL, Xu H, Long X, Overeem S, van Dijk JP, Mischi M: Singular Value Decomposition for Removal of Cardiac Interference from Trunk Electromyogram. Sensors 2021, 21(2):573. [10] Sulistyawan IGE, Nishimae D, Ishii T, Saijo Y: Singular value decomposition with weighting matrix applied for optical-resolution photoacoustic microscopes. Ultrasonics 2024, 143:107424. [11] Krishnan M, Dutta D: A Study of Effectiveness of Principal Component Analysis on Different Data Sets. In: 2017 IEEE International Conference on Computational Intelligence and Computing Research (ICCIC): 14-16 Dec. 2017 2017; 2017: 1-6. [12] Schmidt M, Rajagopal S, Ren Z, Moffat K: Application of singular value decomposition to the analysis of timeresolved macromolecular x-ray data. Biophys J 2003, 84(3):2112-2129.