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Tensor rank problem and multivariate analysis of tensor normal distributions

Research Project

Project/Area Number 24540131
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyushu University

Principal Investigator

Toshio Sakata  九州大学, 芸術工学研究科(研究院), 教授 (20117352)

Co-Investigator(Kenkyū-buntansha) SUMI Toshio  九州大学, 基幹教育研究院, 准教授 (50258513)
MIYAZAKI Mitsuhiro  京都教育大学, 教育学部, 准教授 (90219767)
Co-Investigator(Renkei-kenkyūsha) SASABUCHI Syoichi  九州大学, 芸術工学研究院 (20128028)
KURIKI Satoshi  統計数理研究所 (90195545)
Project Period (FY) 2012-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords3-tensors / typical rank / bilinear forms / determinantal ideal / matrix normal / similar test / テンソルデータの階数 / テンソルデータの典型階数 / 絶対列充足階数テンソル / 正則な双線形形式 / 行列・テンソル型正規分布の平均の片側検定 / テンソルの階数 / テンソルの典型階数 / テンソル正規分布 / 行列正規分布の片側検定 / テンソル正規分布の片側検定 / 制約ホロノミック勾配法 / 2x2x...x2テンソルの階数 / テンソル多変量解析 / Rによるグレブナ基定 / 計算代数統計 / 量子通信と階数
Outline of Final Research Achievements

In statistics, a tensor means a multi-way array and it is an extenstion of matrix. To grasp the meanning of the datum, we decompose the tensor into a sum of rank one tensors, where rank one tensors are the most simple tensors. The rank of a tensor T is the mimimum length of such sum of rank one tensors to express T. A tensor rank is called a typical rank if the tensors with the rank has a positive measure. From a view of data analysis, typical rank is an important concept. In this study, we considered the typical ranks of (m,n,p) type 3-tensors. By using the concepts of absolutely nonsingular tensor, absolutely full column rank tensors and nonsingular bilinear form,determinantal ideal, we decided partially whether there is one typical rank or there are plural typical ranks over the real number field. Also we studied and constructed the one-sided similar test for the mean matrix of matrix type normal distributions.

Report

(5 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (41 results)

All 2016 2015 2014 2013 2012 Other

All Journal Article (9 results) (of which Peer Reviewed: 8 results) Presentation (28 results) (of which Int'l Joint Research: 7 results,  Invited: 3 results) Book (2 results) Remarks (2 results)

  • [Journal Article] On the real radical of certain determinantal ideals defined by tensors2016

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata
    • Journal Title

      第37回可換環論シンポジウム報告集

      Volume: 1 Pages: 121-131

    • Related Report
      2015 Annual Research Report
  • [Journal Article] テンソルの階数問題について (計算代数的視点から)2015

    • Author(s)
      坂田 年男,角 俊雄,宮崎 充弘,前原 貴憲
    • Journal Title

      日本統計学会誌

      Volume: 44(2) Pages: 409-450

    • NAID

      110009930637

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Typical rank of mxnx(m-1)n tensors with 3<=m<=n over the real number field2015

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki and Toshio Sakata
    • Journal Title

      Linear and Multilinear Algebra

      Volume: 63(5) Issue: 5 Pages: 940-955

    • DOI

      10.1080/03081087.2014.910206

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Rank of tensors with size 2 x ... x 22014

    • Author(s)
      Toshio Sumi, Toshio Sakata and Mitsuhiro Miyazaki
    • Journal Title

      Far East J. Math. Sci.

      Volume: 90(2) Pages: 141-162

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Typical ranks of mxnxn(m-1) tensors with 3<=m<=n over the real number field2014

    • Author(s)
      Toshio Sumi, Mitushiro Miyazaki,Toshio Sakata
    • Journal Title

      Linear and Multilinear Algebra

      Volume: 印刷中

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] One-Sided Tests for Matrix Variate Normal Distributins2013

    • Author(s)
      Manabu Iwasa , Toshio Sakata
    • Journal Title

      Proceedings of MSAST2013

      Volume: 2 Pages: 79-91

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Typical ranks for mxnx(m-1)n tensors with m <= n2013

    • Author(s)
      Toshio Sumi, Toshio Sakata and Mitsuhiro Miyazaki.
    • Journal Title

      Linear Algebra and its Application

      Volume: 438 Pages: 953-958

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] An algorithm of finding 4 x 4 x3 absolutely nonsingular tensors based on the Cayley Form of Orthogonal matrices.2012

    • Author(s)
      Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki and Kazumitsu Maehara
    • Journal Title

      , Contemporary Topics in Mathematics and Statistics with Applications, Asian Books Private Limited.

      Volume: 1 Pages: 199-210

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Covariance Estimation Under Tensor Normal Distributions2012

    • Author(s)
      Toshio Sakata, Manabu Iwasa, Teki Go, Reiko Sakurai and Kartina Nurjanni
    • Journal Title

      , E-proceedings of 6th international conference of IMBIC on Mathematical Sciences for Advancement of Science and Technology

      Volume: 1 Pages: 134-148

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] Typical ranks of tensors over the real number field and determinantal ideals2016

    • Author(s)
      Mitsuhiro Miyazaki, Toshio Sumi, Toshio Sakata
    • Organizer
      The 4th Institute of Mathematical Statistics Asia Pacific Rim Meeting
    • Place of Presentation
      The Chinese University of Hong Kong
    • Year and Date
      2016-06-27
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Determinantal varieties and typical ranks of 3-tensors2016

    • Author(s)
      Mitsuhiro Miyazaki, Toshio Sumi, Toshio Sakata
    • Organizer
      Application of Algebraic Methods to Statistics
    • Place of Presentation
      RIMS at Kyoto University
    • Year and Date
      2016-06-20
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the real radical of certain determinantal ideals and multiple typical ranks of 3-tensors over the real number field Ha Long-Vietnam, 2016年3月24日(招待講演)2016

    • Author(s)
      宮崎充弘、角俊雄、坂田年男
    • Organizer
      International Conference and 8th Japan-Vietnam joint Seminar on Commutative Algebra
    • Place of Presentation
      Ha Long-Vietnam
    • Year and Date
      2016-03-21
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] ある種のテンソルにより定義される行列の行列式イデアルの実根基について2016

    • Author(s)
      宮崎充弘、角俊雄、坂田年男
    • Organizer
      日本数学会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-15
    • Related Report
      2015 Annual Research Report
  • [Presentation] On the real radical of certain determinantal ideals defined by tensors2015

    • Author(s)
      宮崎充弘、角俊雄、坂田年男
    • Organizer
      第37回可換環論シンポジウム
    • Place of Presentation
      倉敷シーサイドホテル(岡山県倉敷市)
    • Year and Date
      2015-11-18
    • Related Report
      2015 Annual Research Report
  • [Presentation] About typical ranks of real tensors2015

    • Author(s)
      Mitsuhiro Miyazaki, Toshio Sumi, Toshio Sakata
    • Organizer
      SIAM Conference on Applied Algebraic Geometry
    • Place of Presentation
      National Institute for Mathematical Sciences, 韓国、Daejeon(大田)
    • Year and Date
      2015-08-03
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Groebner Basis Calculation over the finite fields GF(p) and it application to tensor rank determination problem2015

    • Author(s)
      Toshio Sakata, Kosuke Okusa, Toshio Sumi and Mitsuhiro Miyazaki
    • Organizer
      IASC Satellite Conference 2015: Statistical Computing for Data Science
    • Place of Presentation
      Atlantico Buzios Convention and Resort, Bujios, Brazil
    • Year and Date
      2015-08-02
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Verification of absolute nonsingularity of 3 tensors of positivity of a single variate characteristic function2015

    • Author(s)
      Toshio Sakata,Toshio Sumi and Mitsuhiro Miyazaki,
    • Organizer
      60-th ISI Statistics Congress
    • Place of Presentation
      Riocentero, Rio de Jyaneiro,Brazil
    • Year and Date
      2015-07-26
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Similar tests in one-sided testing problems for matrix variate normal distributions2015

    • Author(s)
      Manabu Iwasa, Toshio Sakata
    • Organizer
      IMS-China International Conference on Statistics and Probability
    • Place of Presentation
      Kunming Yun-An Auditrium Hotel, Yunnan, China
    • Year and Date
      2015-06-01
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Actions of special linear groups to tensors of indeterminates and sagbi bases of the rings of invariants2015

    • Author(s)
      宮崎充弘
    • Organizer
      日本数学会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21
    • Related Report
      2014 Research-status Report
  • [Presentation] Monomial preorder, ideals of minors, and plural typical ranks2015

    • Author(s)
      Toshio Sumi, Mitsuhiro MIyazaki, Toshio Sakata
    • Organizer
      Workshop on computational and algebraic methods in statistics
    • Place of Presentation
      東京大学 山上会館
    • Year and Date
      2015-03-03 – 2015-03-05
    • Related Report
      2014 Research-status Report
  • [Presentation] Tensor of indeterminates への群作用とその不変式環のinitial algebra について2015

    • Author(s)
      宮崎充弘
    • Organizer
      第27回可換環論セミナー
    • Place of Presentation
      静岡大学
    • Year and Date
      2015-01-28
    • Related Report
      2014 Research-status Report
  • [Presentation] Group action on tensors of indeterminates and sagbi bases2014

    • Author(s)
      Mitshihiro Miyazaki
    • Organizer
      可換環論と表現論 吉野雄二先生還暦記念研究集会
    • Place of Presentation
      岡山大学
    • Year and Date
      2014-10-13
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Tensor Ranks from Computational Algebraic Methods2014

    • Author(s)
      Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki, Takanori Maehara
    • Organizer
      Algebraic Statistics
    • Place of Presentation
      NIMS, Daejon, South Korea
    • Year and Date
      2014-07-14 – 2014-07-17
    • Related Report
      2014 Research-status Report
  • [Presentation] One-sided tests for matrix and tensor variate normal distributions2014

    • Author(s)
      Manabu Iwasa and Toshio Sakata
    • Organizer
      The 3rd Institute of Mathematical Statistics and Asia Pacific Rim Meeting
    • Place of Presentation
      Howard International House, Taipei, Taiwan
    • Year and Date
      2014-06-29 – 2014-07-04
    • Related Report
      2014 Research-status Report
  • [Presentation] Typical ranks of 3-tensors over the real number field and Hurwitz-Radon function2014

    • Author(s)
      Mitsuhiro Miyazaki,Toshio Sumi,Toshio Sakata
    • Organizer
      The 3rd Institute of Mathematical Statistics and Asia Pacific Rim Meeting
    • Place of Presentation
      Howard International House, Taipei, Taiwan
    • Year and Date
      2014-06-29 – 2014-07-04
    • Related Report
      2014 Research-status Report
  • [Presentation] Rank of n-Tensors with Size 2×・ ・ ・×22014

    • Author(s)
      Toshio Sumi, Toshio Sakata, Mitsuhiro Miyazaki
    • Organizer
      Innovation in statistics and related mathematics through computational algebraic statistics
    • Place of Presentation
      京都テルサ、京都
    • Related Report
      2013 Research-status Report
  • [Presentation] Hurwitz-Radon function, absolutely nonsingular tensor and typical ranks of 3-tensors2014

    • Author(s)
      Mitsuhiro Miyazaki, Toshio Sumi, Toshio Sakata
    • Organizer
      Innovation in statistics and related mathematics through computational algebraic statistics
    • Place of Presentation
      京都テルサ、京都
    • Related Report
      2013 Research-status Report
  • [Presentation] テンソルデータの階数問題(計算代数統計的視点から)2014

    • Author(s)
      坂田年男、角俊雄、宮崎充弘、前原貴憲
    • Organizer
      日本統計学会春季大会
    • Place of Presentation
      同志社大学
    • Related Report
      2013 Research-status Report
  • [Presentation] Holonoic Decent Method for Generalized Fisher Distribution2013

    • Author(s)
      Toshio Sakata, Toshio Sumi, Rieko Sakurai
    • Organizer
      Seventh Annual International Conference on Mathematics Education & Statistics Education, Mathematics and Statistics
    • Place of Presentation
      Athen, Greece
    • Related Report
      2013 Research-status Report
  • [Presentation] Holonomic Decent Minimization Method for Restricted Maximum Likelihood Estimation2013

    • Author(s)
      Rieko Sakurai , and Toshio Sakata
    • Organizer
      59th world congress of the I.S.I
    • Place of Presentation
      Hong Kong Convention and Exhibition Center, Hong Kong
    • Related Report
      2013 Research-status Report
  • [Presentation] Estimation of the typical ranks of 2 x2x...x2 tensors through intrinsic dimension estimators2013

    • Author(s)
      Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    • Organizer
      6th International Conference of the ERCIM WG on Computational and Methodological Statistics
    • Place of Presentation
      Senate house, University of London, UK
    • Related Report
      2013 Research-status Report
  • [Presentation] One-Sided Tests for Matrix Variate Normal Distributions2013

    • Author(s)
      Manabu Iwasa, Toshio Sakata
    • Organizer
      7th International IMBIC Conference MSAST2013
    • Place of Presentation
      INDISMART HOTEL,KOLKATA, INDIA
    • Related Report
      2013 Research-status Report
  • [Presentation] Absolutely full column rank tensors and typical ranks of tensors2012

    • Author(s)
      Mitsuhiro Miyazaki, Toshio Sumi and Toshio Sakata
    • Organizer
      ims-APRM2012
    • Place of Presentation
      つくば国際会議場
    • Related Report
      2012 Research-status Report
  • [Presentation] , A comparison of estimation methods of the Kronecker-delta type covariance matrices of a tensor normal distribution2012

    • Author(s)
      Toshio Sakata,Manabu Iwasa, Teki GO, Rieko Sakurai and Katrina Puji Nurjannni
    • Organizer
      8-th world congress in probability and statistics
    • Place of Presentation
      Istanbul, Torko
    • Related Report
      2012 Research-status Report
  • [Presentation] A Programming to Calculate a Groebner Basis with R2012

    • Author(s)
      Toshio Sakata, Rieko Sakrai, Manabu Iwasa, Toshio Sumi and Mitsuhiro Miyazaki
    • Organizer
      6th International Conference of the ERCIM WORKING Group on Computing and Statistics
    • Place of Presentation
      Oviedo, Spain
    • Related Report
      2012 Research-status Report
  • [Presentation] Covariance Estimation Under Tensor Normal Distributions2012

    • Author(s)
      Toshio Sakata, Manabu Iwasa, Teki Go, Reiko Sakurai and Kartina Nurjanni
    • Organizer
      6th international conference of IMBIC on Mathematical Sciences for Advancement of Science and Technology
    • Place of Presentation
      Kolkata, India
    • Related Report
      2012 Research-status Report
  • [Presentation] 計算機代数統計学のためのRパッケージ:SGROEBNERの開発2012

    • Author(s)
      坂田年男,櫻井利恵子,岩佐学,角俊雄,宮崎充弘
    • Organizer
      日本計算機統計学会第26回シンポジュム
    • Place of Presentation
      東京大学山上会館
    • Related Report
      2012 Research-status Report
  • [Book] Algebraic and Computational Aspects of Real Tensor Ranks2016

    • Author(s)
      Toshio Sakata, Toshio Sumi and Mitsuhiro Miyazaki
    • Total Pages
      121
    • Publisher
      Springer Nature
    • Related Report
      2015 Annual Research Report
  • [Book] Applied Matrix and Tensor Variate Data Analysys2016

    • Author(s)
      Kohei Adachi, Hirokazu Kameoka, Kohei Inoue, Noboru Murata, Deniz Adkdemir, Manabu Iwasa, Toshio Sakata(Toshio Sakata editor)
    • Total Pages
      136
    • Publisher
      Springer Nature
    • Related Report
      2015 Annual Research Report
  • [Remarks] 九州大学 研究者情報

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K002287/index.html

    • Related Report
      2014 Research-status Report
  • [Remarks] 九州大学研究者情報公式ページ

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K002287/index.html

    • Related Report
      2012 Research-status Report

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Published: 2013-05-31   Modified: 2019-07-29  

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