Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible e...

Full description

Saved in:
Bibliographic Details
Published inEntropy (Basel, Switzerland) Vol. 23; no. 9; p. 1214
Main Authors Luo, Yihao, Zhang, Shiqiang, Cao, Yueqi, Sun, Huafei
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 14.09.2021
MDPI
Subjects
Algorithms
Artificial intelligence
Curvature
Data processing
Decomposition
Eigenvalues
Geodesy
Geometry
image edge detecting
Mathematical analysis
Matrices (mathematics)
point cloud denoising
symmetric positive-definite matrix
Symmetry
Wasserstein metric
Online AccessGet full text

Cover

Abstract The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.
AbstractList The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on S P D ( n ) , we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on S P D ( n ) . The experimental results show the efficiency and robustness of our curvature-based methods.
The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.
The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.
Author Sun, Huafei
Luo, Yihao
Zhang, Shiqiang
Cao, Yueqi
AuthorAffiliation 2 Department of Computing, Imperial College London, London SW7 2AZ, UK; s.zhang21@imperial.ac.uk
1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China; knowthingless@bit.edu.cn
3 Department of Mathematics, Imperial College London, London SW7 2AZ, UK; y.cao21@imperial.ac.uk
AuthorAffiliation_xml – name: 1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China; knowthingless@bit.edu.cn
– name: 2 Department of Computing, Imperial College London, London SW7 2AZ, UK; s.zhang21@imperial.ac.uk
– name: 3 Department of Mathematics, Imperial College London, London SW7 2AZ, UK; y.cao21@imperial.ac.uk
Author_xml – sequence: 1
  givenname: Yihao
  surname: Luo
  fullname: Luo, Yihao
– sequence: 2
  givenname: Shiqiang
  surname: Zhang
  fullname: Zhang, Shiqiang
– sequence: 3
  givenname: Yueqi
  surname: Cao
  fullname: Cao, Yueqi
– sequence: 4
  givenname: Huafei
  surname: Sun
  fullname: Sun, Huafei
BookMark eNptkt9rFDEQgINU7A998D8I-NI-nM0ms5vkRShXrQcVCyo-hmwye5djLzmTnOB_716vFFt8ypD55mMmmVNyFFNEQt427L0Qml0iF0w3vIEX5KRhWs9AMHb0T3xMTktZM8YFb7pX5FhAK4US-oTgDaYN1hwcna9stq5iDqUGV2gaaF0h_WlLwVwqhki_HMgU6be76_N4QW30dFELvdpux-BsDSmWffraVkvvcnJYSojL1-TlYMeCbx7OM_Lj08fv88-z2683i_nV7cwBdHVmW94IB36ahEkm9cDcAIMELxUHD9D0UvPOqkY633nmNEfVacGhtVpp2Yszsjh4fbJrs81hY_Mfk2ww9xcpL43N03AjGt5LgF62KISClg_K95YrJjkI6VG4yfXh4Nru-g16h7FmOz6RPs3EsDLL9Nso6CRTMAnOHwQ5_dphqWYTisNxtBHTrhjeSglt17Z8Qt89Q9dpl-P0VHuqAyY47ybq4kC5nErJODw20zCzXwTzuAgTe_mMdaHe_8_Uaxj_U_EX0KOy9Q
CitedBy_id crossref_primary_10_3390_e24101450
Cites_doi 10.1162/089976601750265036
10.1007/978-1-4757-2201-7
10.1109/ICSIDP47821.2019.9173017
10.1016/j.neucom.2004.11.035
10.1109/ICRA.2011.5980567
10.1109/JSTSP.2013.2264798
10.1109/JSTSP.2013.2260320
10.1016/j.sigpro.2021.108176
10.1016/S0167-7152(02)00105-0
10.1007/s41884-018-0014-4
10.1109/TSP.2021.3095725
10.1080/0020739910220413
10.1007/978-0-387-21752-9
10.1007/978-1-4612-0653-8
10.1515/9781400827787
10.1007/s11263-005-3222-z
10.1007/978-3-319-13467-3
10.1137/18M1231389
10.1007/0-387-22726-1_7
ContentType Journal Article
Copyright 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
2021 by the authors. 2021
Copyright_xml – notice: 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
– notice: 2021 by the authors. 2021
DBID AAYXX
CITATION
7TB
8FD
8FE
8FG
ABJCF
ABUWG
AFKRA
AZQEC
BENPR
BGLVJ
CCPQU
DWQXO
FR3
HCIFZ
KR7
L6V
M7S
PHGZM
PHGZT
PIMPY
PKEHL
PQEST
PQGLB
PQQKQ
PQUKI
PRINS
PTHSS
7X8
5PM
DOA
DOI 10.3390/e23091214
DatabaseName CrossRef
Mechanical & Transportation Engineering Abstracts
Technology Research Database
ProQuest SciTech Collection
ProQuest Technology Collection
Materials Science & Engineering Collection
ProQuest Central (Alumni)
ProQuest Central UK/Ireland
ProQuest Central Essentials
ProQuest Central
Technology Collection
ProQuest One
ProQuest Central Korea
Engineering Research Database
SciTech Premium Collection
Civil Engineering Abstracts
ProQuest Engineering Collection
Engineering Database
ProQuest Central Premium
ProQuest One Academic (New)
Publicly Available Content Database
ProQuest One Academic Middle East (New)
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Applied & Life Sciences
ProQuest One Academic
ProQuest One Academic UKI Edition
ProQuest Central China
Engineering Collection
MEDLINE - Academic
PubMed Central (Full Participant titles)
DOAJ Directory of Open Access Journals
DatabaseTitle CrossRef
Publicly Available Content Database
Technology Collection
Technology Research Database
ProQuest One Academic Middle East (New)
Mechanical & Transportation Engineering Abstracts
ProQuest Central Essentials
ProQuest Central (Alumni Edition)
SciTech Premium Collection
ProQuest One Community College
ProQuest Central China
ProQuest Central
ProQuest One Applied & Life Sciences
ProQuest Engineering Collection
ProQuest Central Korea
ProQuest Central (New)
Engineering Collection
Civil Engineering Abstracts
Engineering Database
ProQuest One Academic Eastern Edition
ProQuest Technology Collection
ProQuest SciTech Collection
ProQuest One Academic UKI Edition
Materials Science & Engineering Collection
Engineering Research Database
ProQuest One Academic
ProQuest One Academic (New)
MEDLINE - Academic
DatabaseTitleList
CrossRef
MEDLINE - Academic

Publicly Available Content Database
Database_xml – sequence: 1
  dbid: DOA
  name: DOAJ Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
– sequence: 2
  dbid: 8FG
  name: ProQuest Technology Collection
  url: https://search.proquest.com/technologycollection1
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
EISSN 1099-4300
ExternalDocumentID oai_doaj_org_article_2b744b75e338452f8dba28072437de3c
PMC8467084
10_3390_e23091214
GroupedDBID 29G
2WC
5GY
5VS
8FE
8FG
AADQD
AAFWJ
AAYXX
ABDBF
ABJCF
ACIWK
ACUHS
ADBBV
AEGXH
AENEX
AFKRA
AFPKN
AFZYC
ALMA_UNASSIGNED_HOLDINGS
BCNDV
BENPR
BGLVJ
CCPQU
CITATION
CS3
DU5
E3Z
ESX
F5P
GROUPED_DOAJ
GX1
HCIFZ
HH5
IAO
ITC
J9A
KQ8
L6V
M7S
MODMG
M~E
OK1
OVT
PGMZT
PHGZM
PHGZT
PIMPY
PROAC
PTHSS
RNS
RPM
TR2
TUS
XSB
~8M
7TB
8FD
ABUWG
AZQEC
DWQXO
FR3
KR7
PKEHL
PQEST
PQGLB
PQQKQ
PQUKI
PRINS
7X8
5PM
PUEGO
ID FETCH-LOGICAL-c446t-a5213c4d21407079f0cf4f74d7824d441b7926a817cd6d0c92e8693245a9897b3
IEDL.DBID DOA
ISSN 1099-4300
IngestDate Wed Aug 27 01:09:19 EDT 2025
Thu Aug 21 18:04:27 EDT 2025
Thu Jul 10 22:10:33 EDT 2025
Fri Jul 25 12:01:20 EDT 2025
Tue Jul 01 01:58:05 EDT 2025
Thu Apr 24 23:06:09 EDT 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 9
Language English
License https://creativecommons.org/licenses/by/4.0
Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c446t-a5213c4d21407079f0cf4f74d7824d441b7926a817cd6d0c92e8693245a9897b3
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
content type line 23
OpenAccessLink https://doaj.org/article/2b744b75e338452f8dba28072437de3c
PMID 34573839
PQID 2576403226
PQPubID 2032401
ParticipantIDs doaj_primary_oai_doaj_org_article_2b744b75e338452f8dba28072437de3c
pubmedcentral_primary_oai_pubmedcentral_nih_gov_8467084
proquest_miscellaneous_2577456552
proquest_journals_2576403226
crossref_primary_10_3390_e23091214
crossref_citationtrail_10_3390_e23091214
ProviderPackageCode CITATION
AAYXX
PublicationCentury 2000
PublicationDate 20210914
PublicationDateYYYYMMDD 2021-09-14
PublicationDate_xml – month: 9
  year: 2021
  text: 20210914
  day: 14
PublicationDecade 2020
PublicationPlace Basel
PublicationPlace_xml – name: Basel
PublicationTitle Entropy (Basel, Switzerland)
PublicationYear 2021
Publisher MDPI AG
MDPI
Publisher_xml – name: MDPI AG
– name: MDPI
References Li (ref_9) 2013; 7
ref_14
Givens (ref_12) 1984; 2
ref_13
ref_10
Arsigny (ref_7) 2010; 2
Fiori (ref_26) 2001; 13
Massart (ref_15) 2020; 1
ref_19
ref_18
ref_16
Nishimori (ref_25) 2005; 67
Nadernejad (ref_24) 2008; 2
Ward (ref_17) 1991; 22
Hua (ref_27) 2021; 69
ref_22
ref_21
ref_20
Masry (ref_23) 2002; 2
ref_1
Gelbrich (ref_11) 2015; 1
ref_3
ref_29
ref_8
Manton (ref_2) 2013; 4
Pennec (ref_6) 2006; 1
ref_5
Hua (ref_28) 2021; 188
ref_4
References_xml – volume: 1
  start-page: 185
  year: 2015
  ident: ref_11
  article-title: On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces
  publication-title: Math. Nachrichten
– ident: ref_3
– volume: 2
  start-page: 411
  year: 2010
  ident: ref_7
  article-title: Log-Euclidean Metrics for Fast and Simple Calculus on Diffusion Tensors
  publication-title: Magn. Reson. Med.
– volume: 13
  start-page: 1625
  year: 2001
  ident: ref_26
  article-title: A Theory for Learning by Weight Flow on Stiefel-Grassman Manifold
  publication-title: Neural Comput.
  doi: 10.1162/089976601750265036
– ident: ref_18
  doi: 10.1007/978-1-4757-2201-7
– ident: ref_10
  doi: 10.1109/ICSIDP47821.2019.9173017
– volume: 67
  start-page: 106
  year: 2005
  ident: ref_25
  article-title: Learning Algorithms Utilizing Quasi-geodesic Flows on the Stiefel Manifold
  publication-title: Neurocomputing
  doi: 10.1016/j.neucom.2004.11.035
– ident: ref_16
  doi: 10.1109/ICRA.2011.5980567
– volume: 4
  start-page: 681
  year: 2013
  ident: ref_2
  article-title: A Primer on Stochastic Differential Geometry for Signal Processing
  publication-title: IEEE J. Sel. Top. Signal Process.
  doi: 10.1109/JSTSP.2013.2264798
– volume: 7
  start-page: 655
  year: 2013
  ident: ref_9
  article-title: Riemannian Distances for Signal Classification by Power Spectral Density
  publication-title: IEEE J. Sel. Top. Signal Process.
  doi: 10.1109/JSTSP.2013.2260320
– ident: ref_14
– volume: 188
  start-page: 108176
  year: 2021
  ident: ref_28
  article-title: MIG Median Detectors with Manifold Filter
  publication-title: Signal Process.
  doi: 10.1016/j.sigpro.2021.108176
– ident: ref_1
– volume: 2
  start-page: 205
  year: 2002
  ident: ref_23
  article-title: Multivariate Probability Density Estimation for Associated Processes: Strong Consistency and Rates
  publication-title: Stat. Probab. Lett.
  doi: 10.1016/S0167-7152(02)00105-0
– volume: 2
  start-page: 1507
  year: 2008
  ident: ref_24
  article-title: Edge Detection Techniques: Evaluations and Comparisons
  publication-title: Appl. Math. Sci.
– ident: ref_29
  doi: 10.1007/s41884-018-0014-4
– volume: 69
  start-page: 4326
  year: 2021
  ident: ref_27
  article-title: Target Detection Within Nonhomogeneous Clutter Via Total Bregman Divergence-Based Matrix Information Geometry Detectors
  publication-title: IEEE Trans. Signal Process.
  doi: 10.1109/TSP.2021.3095725
– volume: 22
  start-page: 615
  year: 1991
  ident: ref_17
  article-title: A General Analysis of Sylvester’s Matrix Equation
  publication-title: Int. J. Math. Educ. Sci. Technol.
  doi: 10.1080/0020739910220413
– ident: ref_19
  doi: 10.1007/978-0-387-21752-9
– ident: ref_4
  doi: 10.1007/978-1-4612-0653-8
– ident: ref_5
  doi: 10.1515/9781400827787
– volume: 1
  start-page: 41
  year: 2006
  ident: ref_6
  article-title: A Riemannian Framework for Tensor Computing
  publication-title: Int. J. Comput. Vis.
  doi: 10.1007/s11263-005-3222-z
– ident: ref_8
  doi: 10.1007/978-3-319-13467-3
– ident: ref_13
– volume: 1
  start-page: 171
  year: 2020
  ident: ref_15
  article-title: Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices
  publication-title: Siam J. Matrix Anal. Appl.
  doi: 10.1137/18M1231389
– ident: ref_22
– ident: ref_20
– ident: ref_21
  doi: 10.1007/0-387-22726-1_7
– volume: 2
  start-page: 231
  year: 1984
  ident: ref_12
  article-title: A Class of Wasserstein Metrics for Probability Distributions
  publication-title: Mich. Math. J.
SSID ssj0023216
Score 2.2645564
Snippet The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence...
SourceID doaj
pubmedcentral
proquest
crossref
SourceType Open Website
Open Access Repository
Aggregation Database
Enrichment Source
Index Database
StartPage 1214
SubjectTerms Algorithms
Artificial intelligence
Curvature
Data processing
Decomposition
Eigenvalues
Geodesy
Geometry
image edge detecting
Mathematical analysis
Matrices (mathematics)
point cloud denoising
symmetric positive-definite matrix
Symmetry
Wasserstein metric
SummonAdditionalLinks – databaseName: ProQuest Central
  dbid: BENPR
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1Nb9YwDI5gu3BBQ4AojClMHMahWpu6TXpC-2RD2jQBE7tViZMCErTb3u7_z-6bt6wS4ppY_XA-_DiOHwvx3mgVCkIWKfrSpVDmLnUOy5RzME1mqUFx7vDZeXVyCZ-vyqt44LaI1ypXe-K4Ufse-Yx8l4ExZDT9qo_XNylXjeLoaiyh8Vis0xZsyPla3z86v_gyuVyFyqsln1BBzv1uIMBd5yqHmRUayfpnCHN-P_KBwTneEE8jUpR7y6F9Jh6F7rkIn0L_h4tgoTyYUy3LvpUE5uR3OyZQchFLebaU7Dv59eJwp_sgbefl6bCQew_C1tx9aAcrY84A2bIX4vL46NvBSRorJaRI7tyQWjLCBYKnPxsp79oMW2g1eLL_4AnxOF2ryppco698hrUKpiLkBqWtTa1d8VKsdX0XXgmZIWIbMtTQ1lApdPSc3AMq0MFYaxKxs9Jcg5FGnKtZ_G7InWAlN5OSE7E9iV4vuTP-JbTP6p8EmO56bOhvfzRx9TTKaQCny0AONZSqNd5ZpvFhNkUfCkzE5mrwmrgGF83fGZOId1M3rR4Oidgu9HejjGZMW6pE6Nmgzz5o3tP9-jnycDN0ywy8_v_L34gniu_BcNkJ2BRrw-1deEtAZnBbcbbeA_7I9Dg
  priority: 102
  providerName: ProQuest
Title Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing
URI https://www.proquest.com/docview/2576403226
https://www.proquest.com/docview/2577456552
https://pubmed.ncbi.nlm.nih.gov/PMC8467084
https://doaj.org/article/2b744b75e338452f8dba28072437de3c
Volume 23
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1La9wwEB6a9JJLSGlD3KaLWnpIDya2PLbkY167aSEhNA3dm9HLJNDaJXH-f2ds77KGQi-9-CANRpqRNN8gzTcAn7SSISNkETuf2xjz1MbWujzmHEydGGqQnDt8dV1c3uHXZb7cKPXFb8IGeuBBccfSKkSr8kCxFOay1t4aZnBhIj0fMsenL_m8VTA1hlqZTIuBRyijoP44ENAuU5nixPv0JP0TZDl9F7nhaOZ7sDsiRHEyjOwVvAjNawiL0P7i4ldOnE0plkVbCwJx4ofpEye5eKW4GiTbRtzenB81n4VpvPjSPYmTjetq7j43nRFjrgD5sDdwN7_4fnYZjxUSYkdhXBcbcr6ZQ08z66nu6sTVWCv05PfRE9KxqpSF0alyvvCJK2XQBSE2zE2pS2Wzfdhu2iYcgEicc3VInMK6xEI6S_9JPTqJKmhjdARHK81VbqQP5yoWPysKI1jJ1VrJEXxci_4eODP-JnTK6l8LMM1130DGr0bjV_8yfgSHK-NV4957qjiEwoQOqiKCD-tu2jV8FWKa0D73MoqxbC4jUBOjTwY07Wke7nv-bYZsica3_2MG72BH8isZLkqBh7DdPT6H9wRzOjuDLT1fzODl6cX1zbdZv77pu1imfwCZ_v4q
linkProvider Directory of Open Access Journals
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lb9QwELZKOcAFgQCxUMAgkMohauJMYueAUOmy3aXdColW9Jb6FVoJktJNhfhT_EZm8qKRELde45ETjWc838Sebxh7paTwMSKLwLrEBJBEJjDGJgHVYKpQ4wNBtcPLg3R-BB-Pk-M19ruvhaFrlf2e2GzUrrL0j3yLgDGEaH7pu_MfAXWNotPVvoVGaxZ7_tdPTNlWbxdTXN_XQsw-HO7Mg66rQGAx9akDjQErtuAEphZED1eEtoBCgsNYCQ7RgZGZSLWKpHWpC20mvEoR5UCiM5VJE-O8N9hNiOOMPErNdocELxZR2rIX4WC45RHeZ5GIYBTzmtYAIzw7vo15JbzN7rI7HS7l260h3WNrvrzP_K6vvlPLLct3xsTOvCo4Qkf-RTflmtQyky9byarknz9NN8s3XJeOL-oV375ySE7DU11r3lUoYOR8wI6uRYMP2XpZlf4R46G1tvChlVBkkAprcJ7IgRUgvdJaTdhmr7ncdqTl1DvjW47JCyk5H5Q8YS8H0fOWqeNfQu9J_YMAkWs3D6qLr3nnq7kwEsDIxGP6DokolDOaSIOIu9H52E7YRr94eefxq_yvfU7Yi2EYfZUOYHTpq8tGRhKCTsSEydGijz5oPFKenTas3wQUQwWP___y5-zW_HC5n-8vDvaesNuCbuBQwwvYYOv1xaV_ihCqNs8au-Xs5Lod5Q-DbC11
linkToPdf http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3da9RAEB_qFcQXUVSMVl1FoT6ESzabbPIg0vZ69qw9DrXYt3S_ooImtZci_mv-dc7kywbEt75mJ5swO5P5TXbnNwDPU8ldhMjCNzbWvohD7WttYp9qMNNA4QVOtcNHy-TgWLw9iU824HdfC0PHKvtvYvOhtpWhf-RTAsYiQPNLpkV3LGI1m78---FTBynaae3babQmcuh-_cT0bf1qMcO1fsH5fP_j3oHfdRjwDaZBta8weEVGWI5pBlHFFYEpRCGFxbgpLCIFLTOeqDSUxiY2MBl3aYKIR8QqSzOpI5z3GmxKunsCm7v7y9X7Id2LeJi0XEZRlAVTh2A_C3koRhGwaRQwQrfjs5mXgt38FtzsUCrbac3qNmy48g64N676Tg24DNsb0zyzqmAIJNkn1RRvUgNNdtRKViX7sJptly-ZKi1b1Gu2c2nLnIZnqlasq1fAOHoXjq9Eh_dgUlaluw8sMMYULjBSFJlIuNE4T2iF4UK6VKnUg-1ec7npKMypk8a3HFMZUnI-KNmDZ4PoWcvb8S-hXVL_IEBU282F6vxz3nluzrUUQsvYYTIvYl6kViuiECImR-si48FWv3h55__r_K-1evB0GEbPpe0YVbrqopGRhKdj7oEcLfrohcYj5dcvDQc4wcYgFQ_-__AncB2dJH-3WB4-hBucjuNQ9wuxBZP6_MI9QjxV68ed4TI4vWpf-QOD7TMH
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Geometric+Characteristics+of+the+Wasserstein+Metric+on+SPD%28n%29+and+Its+Applications+on+Data+Processing&rft.jtitle=Entropy+%28Basel%2C+Switzerland%29&rft.au=Luo%2C+Yihao&rft.au=Zhang%2C+Shiqiang&rft.au=Cao%2C+Yueqi&rft.au=Sun%2C+Huafei&rft.date=2021-09-14&rft.issn=1099-4300&rft.eissn=1099-4300&rft.volume=23&rft.issue=9&rft_id=info:doi/10.3390%2Fe23091214&rft.externalDBID=NO_FULL_TEXT
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1099-4300&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1099-4300&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1099-4300&client=summon