Editing Singular value decomposition (section)

===Other examples===
The SVD is also applied extensively to the study of linear [[inverse problem]]s and is useful in the analysis of regularization methods such as that of [[Tikhonov regularization|Tikhonov]]. It is widely used in statistics, where it is related to [[principal component analysis]] and to [[correspondence analysis]], and in [[signal processing]] and [[pattern recognition]]. It is also used in output-only [[modal analysis]], where the non-scaled [[mode shape]]s can be determined from the singular vectors. Yet another usage is [[latent semantic indexing]] in natural-language text processing.

In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" <math>\kappa := \sigma_\text{max} / \sigma_\text{min}</math>. It often controls the error rate or convergence rate of a given computational scheme on such systems.<ref>{{cite journal |last1=Edelman |first1=Alan |title=On the distribution of a scaled condition number |url=http://math.mit.edu/~edelman/publications/distribution_of_a_scaled.pdf |journal=Math. Comp. |volume=58 |pages=185–190 |year=1992|issue=197 |doi=10.1090/S0025-5718-1992-1106966-2 |bibcode=1992MaCom..58..185E |doi-access=free }}</ref><ref>{{cite journal |last1=Shen |first1=Jianhong (Jackie) |title=On the singular values of Gaussian random matrices |journal=Linear Alg. Appl. |volume=326 |pages=1–14 |year=2001|issue=1–3 |doi=10.1016/S0024-3795(00)00322-0 |doi-access=free }}</ref>

The SVD also plays a crucial role in the field of [[quantum information]], in a form often referred to as the [[Schmidt decomposition]]. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be [[Quantum entanglement|entangled]]: if the rank of the <math>\mathbf \Sigma</math> matrix is larger than one.

One application of SVD to rather large matrices is in [[numerical weather prediction]], where [[Lanczos algorithm|Lanczos methods]] are used to estimate the most linearly quickly growing few perturbations to the central numerical weather prediction over a given initial forward time period; i.e., the singular vectors corresponding to the largest singular values of the linearized propagator for the global weather over that time interval. The output singular vectors in this case are entire weather systems. These perturbations are then run through the full nonlinear model to generate an [[ensemble forecasting|ensemble forecast]], giving a handle on some of the uncertainty that should be allowed for around the current central prediction.

SVD has also been applied to reduced order modelling. The aim of reduced order modelling is to reduce the number of degrees of freedom in a complex system which is to be modeled. SVD was coupled with [[radial basis functions]] to interpolate solutions to three-dimensional unsteady flow problems.<ref>{{cite journal | last1 = Walton | first1 = S. | last2 = Hassan | first2 = O. | last3 = Morgan | first3 = K. | year = 2013| title = Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions | journal = Applied Mathematical Modelling | volume =  37| issue = 20–21| pages = 8930–8945| doi=10.1016/j.apm.2013.04.025| doi-access = free }}</ref>

Interestingly, SVD has been used to improve gravitational waveform modeling by the ground-based gravitational-wave interferometer aLIGO.<ref>{{cite journal | last1 = Setyawati | first1 = Y. | last2 = Ohme | first2 = F. | last3 = Khan | first3 = S. | year = 2019| title = Enhancing gravitational waveform model through dynamic calibration | journal = Physical Review D | volume =  99| issue =2 | pages = 024010| doi=10.1103/PhysRevD.99.024010| bibcode = 2019PhRvD..99b4010S | arxiv = 1810.07060 | s2cid = 118935941 }}</ref> SVD can help to increase the accuracy and speed of waveform generation to support gravitational-waves searches and update two different waveform models.

Singular value decomposition is used in [[recommender systems]] to predict people's item ratings.<ref>{{cite report |last1=Sarwar |first1=Badrul |last2=Karypis |first2=George |last3=Konstan |first3=Joseph A. |author3-link=Joseph A. Konstan |last4=Riedl |first4=John T. |author4-link=John T. Riedl |name-list-style=amp |year=2000 |title=Application of Dimensionality Reduction in Recommender System – A Case Study |url=https://apps.dtic.mil/sti/citations/tr/ADA439541 |journal= |publisher=[[University of Minnesota]]|hdl=11299/215429|type=Technical report 00-043}}</ref> Distributed algorithms have been developed for the purpose of calculating the SVD on clusters of commodity machines.<ref>{{cite arXiv |last1=Bosagh Zadeh |first1=Reza |last2=Carlsson |first2=Gunnar |title=Dimension Independent Matrix Square Using MapReduce |year=2013 |class=cs.DS |eprint=1304.1467}}</ref>

Low-rank SVD has been applied for hotspot detection from spatiotemporal data with application to disease [[outbreak]] detection.<ref>{{Cite journal
 |author1=Hadi Fanaee Tork |author2=João Gama
 |title = Eigenspace method for spatiotemporal hotspot detection
 |journal = Expert Systems
 |volume=32
 |issue=3
 |pages = 454–464
 |date = September 2014
 |doi = 10.1111/exsy.12088
 |arxiv=1406.3506 |bibcode=2014arXiv1406.3506F
 |s2cid=15476557
 }}</ref> A combination of SVD and [[Higher-order singular value decomposition|higher-order SVD]] also has been applied for real time event detection from complex data streams (multivariate data with space and time dimensions) in [[disease surveillance]].<ref>{{Cite journal
 |author1=Hadi Fanaee Tork |author2=João Gama
 |title = EigenEvent: An Algorithm for Event Detection from Complex Data Streams in Syndromic Surveillance
 |journal = Intelligent Data Analysis
 |volume = 19
 |issue = 3
 |pages=597–616
 |date = May 2015
 |arxiv = 1406.3496
 |doi=10.3233/IDA-150734|s2cid=17966555
 }}</ref>

In [[astrodynamics]], the SVD and its variants are used as an option to determine suitable maneuver directions for transfer trajectory design<ref name=muralidharan2023stretching>{{Cite journal|title=Stretching directions in cislunar space: Applications for departures and transfer design|first1=Vivek|last1=Muralidharan|first2=Kathleen|last2=Howell|journal =Astrodynamics  | volume = 7 | issue = 2| pages = 153–178 | date = 2023 | doi = 10.1007/s42064-022-0147-z | bibcode = 2023AsDyn...7..153M|s2cid=252637213 }}</ref> and [[orbital station-keeping]].<ref name=Muralidharan2021>{{Cite journal|title=Leveraging stretching directions for stationkeeping in Earth-Moon halo orbits |first1=Vivek|last1=Muralidharan|first2=Kathleen|last2=Howell|journal =[[Advances in Space Research]]  | volume = 69 | issue = 1| pages = 620–646 | date = 2022 | doi = 10.1016/j.asr.2021.10.028 | bibcode = 2022AdSpR..69..620M|s2cid=239490016 }}</ref>

The SVD can be used to measure the similarity between real-valued matrices.<ref name=albers2025>{{Cite journal|title=Assessing the Similarity of Real Matrices with Arbitrary Shape|first1=Jasper|last1=Albers|first2=Anno|last2=Kurth|first3=Robin|last3=Gutzen|first4=Aitor|last4=Morales-Gregorio|first5=Michael|last5=Denker|first6=Sonja|last6=Gruen|first7=Sacha|last7=van Albada|first8=Markus|last8=Diesmann|journal=PRX Life| issue = 3| pages = 023005 | date = 2025 | doi = 10.1103/PRXLife.3.023005 |arxiv=2403.17687}}</ref> By measuring the angles between the singular vectors, the inherent two-dimensional structure of matrices is accounted for. This method was shown to outperform [[cosine similarity]] and [[Frobenius norm]] in most cases, including brain activity measurements from [[neuroscience]] experiments.