![Page 1: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/1.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Aspects of Group Theory in Stochastic Problems
Dr. Marconi BarbosaNICTA/ANU, Canberra, Australia
November 18, 2008
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 2: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/2.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 3: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/3.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 4: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/4.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 5: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/5.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 6: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/6.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 7: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/7.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 8: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/8.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Mallow’s Model
I Harmonic analysis on manifolds
I Fourier transforms on groups
I Graph matching: edge info added into node features
I What we do know about Metropolis algorithm? ExactResults?
I Fastest Mixing Markov Chains
I Parallel Coset enumeration
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 9: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/9.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 10: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/10.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 11: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/11.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 12: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/12.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 13: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/13.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 14: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/14.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 15: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/15.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 16: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/16.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 17: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/17.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 18: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/18.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 19: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/19.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 20: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/20.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
Outline of what is (would be nice) to come...
I Lagrange Theorem
I Example: Fermat Little theorem and cryptography
I Orbit Counting Theorem
I Example: Cube orbits
I Magic cube group: Scary
I More scary: Baby Monster
I Freaking out: The Monster group
I Group classification: one slide soft crash course
I Group representation: one slide hard crash course
I Invariance, equivalence and symmetry
I Differential invariants, variational problems with symmetry.
I Geometric probability, Minkowski functionals and continuousGroups.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 21: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/21.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Verducci & Fligner: Distance Based Ranking Models
I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.
I Fagin: Comparing partial ranks, how efficient?
I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)
I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)
I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 22: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/22.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Verducci & Fligner: Distance Based Ranking Models
I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.
I Fagin: Comparing partial ranks, how efficient?
I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)
I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)
I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 23: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/23.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Verducci & Fligner: Distance Based Ranking Models
I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.
I Fagin: Comparing partial ranks, how efficient?
I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)
I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)
I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 24: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/24.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Verducci & Fligner: Distance Based Ranking Models
I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.
I Fagin: Comparing partial ranks, how efficient?
I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)
I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)
I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 25: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/25.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Verducci & Fligner: Distance Based Ranking Models
I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.
I Fagin: Comparing partial ranks, how efficient?
I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)
I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)
I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 26: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/26.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
SceneFinite Groups crash course outlineMotivational Papers
I Verducci & Fligner: Distance Based Ranking Models
I Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain withsymmetry, 2006.
I Fagin: Comparing partial ranks, how efficient?
I Guy Lebannon: Partial Rankings and Cosets/Posets(nips2007)
I Risi Kondor:Multi-Object tracking...with...simmetric group (nips2006)Diffusion Kernel in graphs and other structures (manifoldmethods)
I J.Huang C. Guestrin & L. Gibbas: Polytope projection afterFourier inverse transform and estimation in Fourier Domain.(nips2007)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 27: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/27.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
A group G is a set S and a binary operation ∗ satisfying twoproperties:
I A) Closure. For a, b ∈ G , then a ∗ b ∈ G .
I B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) arethe same.
There must be two very special members too:
I C) The identity element e is such that:a ∗ e = e ∗ a = a
I D) The inverse b, for any member a:a ∗ b = b ∗ a = e
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 28: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/28.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
A group G is a set S and a binary operation ∗ satisfying twoproperties:
I A) Closure. For a, b ∈ G , then a ∗ b ∈ G .
I B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) arethe same.
There must be two very special members too:
I C) The identity element e is such that:a ∗ e = e ∗ a = a
I D) The inverse b, for any member a:a ∗ b = b ∗ a = e
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 29: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/29.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
A group G is a set S and a binary operation ∗ satisfying twoproperties:
I A) Closure. For a, b ∈ G , then a ∗ b ∈ G .
I B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) arethe same.
There must be two very special members too:
I C) The identity element e is such that:a ∗ e = e ∗ a = a
I D) The inverse b, for any member a:a ∗ b = b ∗ a = e
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 30: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/30.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
A group G is a set S and a binary operation ∗ satisfying twoproperties:
I A) Closure. For a, b ∈ G , then a ∗ b ∈ G .
I B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) arethe same.
There must be two very special members too:
I C) The identity element e is such that:a ∗ e = e ∗ a = a
I D) The inverse b, for any member a:a ∗ b = b ∗ a = e
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 31: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/31.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
A group G is a set S and a binary operation ∗ satisfying twoproperties:
I A) Closure. For a, b ∈ G , then a ∗ b ∈ G .
I B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) arethe same.
There must be two very special members too:
I C) The identity element e is such that:a ∗ e = e ∗ a = a
I D) The inverse b, for any member a:a ∗ b = b ∗ a = e
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 32: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/32.jpg)
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I A subgroup of G , is any subset of elements that still forms agroup.
I Consider a subgroup H of G . Associated with H we createthe following setgH = {g .h : ∀h ∈ H}, This is one of the left cosets of H inG , indexed by g .
Next: An example from integers, the Zn family.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 33: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/33.jpg)
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I A subgroup of G , is any subset of elements that still forms agroup.
I Consider a subgroup H of G . Associated with H we createthe following setgH = {g .h : ∀h ∈ H}, This is one of the left cosets of H inG , indexed by g .
Next: An example from integers, the Zn family.
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 34: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/34.jpg)
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Coset example from naturals
Group: G = Z4
Set S : {0, 1, 2, 3}Operation ∗: integer addition mod 4Subgroup: H = {0, 2}0 ∗ H = {0, 2} = H, a trivial coset.1 ∗ H = {1, 3}, first one here2 ∗ H = {2, 4} = {2, 0} = H, trivial again3 ∗ H = {3, 5} = {3, 1}, nothing new...So the subgroup H has only two cosets:H itself and {1, 3}.Note that the cosets form a partition of the group:Z4 = (1 ∗ H)
⋃H
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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The order of every subgroup H divides the order of G .
I Show that all cosets of H have the same order. Definef : aH ⇒ bH by f = ba−1. This is a bijective map with inversef −1 = ab−1.
I Show that cosets from H form a partition of G : Cosets are eitheridentical or disjoint: every element belongs to only one coset.
I Then, the number of elements in G is equal the number of cosets(index) times the number of elements in each coset (which inturn is equal to the order of H)
|G | = |union of its H-cosets|= (number of cosets)*(number of elements in a coset)
= |G : H|.|H|
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
The order of every subgroup H divides the order of G .
I Show that all cosets of H have the same order. Definef : aH ⇒ bH by f = ba−1. This is a bijective map with inversef −1 = ab−1.
I Show that cosets from H form a partition of G : Cosets are eitheridentical or disjoint: every element belongs to only one coset.
I Then, the number of elements in G is equal the number of cosets(index) times the number of elements in each coset (which inturn is equal to the order of H)
|G | = |union of its H-cosets|= (number of cosets)*(number of elements in a coset)
= |G : H|.|H|
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 37: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/37.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
The order of every subgroup H divides the order of G .
I Show that all cosets of H have the same order. Definef : aH ⇒ bH by f = ba−1. This is a bijective map with inversef −1 = ab−1.
I Show that cosets from H form a partition of G : Cosets are eitheridentical or disjoint: every element belongs to only one coset.
I Then, the number of elements in G is equal the number of cosets(index) times the number of elements in each coset (which inturn is equal to the order of H)
|G | = |union of its H-cosets|= (number of cosets)*(number of elements in a coset)
= |G : H|.|H|
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 38: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/38.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
The order of every subgroup H divides the order of G .
I Show that all cosets of H have the same order. Definef : aH ⇒ bH by f = ba−1. This is a bijective map with inversef −1 = ab−1.
I Show that cosets from H form a partition of G : Cosets are eitheridentical or disjoint: every element belongs to only one coset.
I Then, the number of elements in G is equal the number of cosets(index) times the number of elements in each coset (which inturn is equal to the order of H)
|G | = |union of its H-cosets|= (number of cosets)*(number of elements in a coset)
= |G : H|.|H|
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 39: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/39.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
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Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Consequences
The order of any element a of a finite group (i.e. the smallest k forwhich ak = e divides G ) divides the order of G . Because the order ofa is the order of a cyclic subgroup generated by a.
I Fermat’s Little theoremap ≡ a (mod p) ;p prime, a integer
I Euler’s Theoremaφ(n) ≡ 1 (mod n); a co-prime nφ(n)(Euler function) counts the number of co-primes from 1 to n.
I Carmichael’s Theorem aλ(n) ≡ 1 (mod n)λ(n)(Carmichael’s function) gives the smallest integer m forwhich a(m) ≡ 1 (mod n)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 40: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/40.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Consequences
The order of any element a of a finite group (i.e. the smallest k forwhich ak = e divides G ) divides the order of G . Because the order ofa is the order of a cyclic subgroup generated by a.
I Fermat’s Little theoremap ≡ a (mod p) ;p prime, a integer
I Euler’s Theoremaφ(n) ≡ 1 (mod n); a co-prime nφ(n)(Euler function) counts the number of co-primes from 1 to n.
I Carmichael’s Theorem aλ(n) ≡ 1 (mod n)λ(n)(Carmichael’s function) gives the smallest integer m forwhich a(m) ≡ 1 (mod n)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 41: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/41.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Consequences
The order of any element a of a finite group (i.e. the smallest k forwhich ak = e divides G ) divides the order of G . Because the order ofa is the order of a cyclic subgroup generated by a.
I Fermat’s Little theoremap ≡ a (mod p) ;p prime, a integer
I Euler’s Theoremaφ(n) ≡ 1 (mod n); a co-prime nφ(n)(Euler function) counts the number of co-primes from 1 to n.
I Carmichael’s Theorem aλ(n) ≡ 1 (mod n)λ(n)(Carmichael’s function) gives the smallest integer m forwhich a(m) ≡ 1 (mod n)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 42: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/42.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Fermat Little theorem proof by group theory
I Basic idea is to recognize G = {1, 2, ..., p − 1}, with theoperation of multiplication mod p as a Group. Some work toprove that every element is invertible
I Assume that a is an element of G and let k be its order. i.e.
I ak ≡ 1 (mod p)
I by Lagrange theorem, k divides the order of G , which is p− 1.So p− 1 = k ∗m then ap−1 = ak∗m = (ak)m = 1m = 1 mod p�
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 43: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/43.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Fermat Little theorem proof by group theory
I Basic idea is to recognize G = {1, 2, ..., p − 1}, with theoperation of multiplication mod p as a Group. Some work toprove that every element is invertible
I Assume that a is an element of G and let k be its order. i.e.
I ak ≡ 1 (mod p)
I by Lagrange theorem, k divides the order of G , which is p− 1.So p− 1 = k ∗m then ap−1 = ak∗m = (ak)m = 1m = 1 mod p�
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 44: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/44.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Fermat Little theorem proof by group theory
I Basic idea is to recognize G = {1, 2, ..., p − 1}, with theoperation of multiplication mod p as a Group. Some work toprove that every element is invertible
I Assume that a is an element of G and let k be its order. i.e.
I ak ≡ 1 (mod p)
I by Lagrange theorem, k divides the order of G , which is p− 1.So p− 1 = k ∗m then ap−1 = ak∗m = (ak)m = 1m = 1 mod p�
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 45: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/45.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Fermat Little theorem proof by group theory
I Basic idea is to recognize G = {1, 2, ..., p − 1}, with theoperation of multiplication mod p as a Group. Some work toprove that every element is invertible
I Assume that a is an element of G and let k be its order. i.e.
I ak ≡ 1 (mod p)
I by Lagrange theorem, k divides the order of G , which is p− 1.So p− 1 = k ∗m then ap−1 = ak∗m = (ak)m = 1m = 1 mod p�
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 46: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/46.jpg)
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Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Fermat Little theorem proof by group theory
I Invertibility property.
I Assume b is co-prime (relative prime) to p. Using Bezoutidentity (a linear Diophantine equation)
I bx + py = 1 ( x , y integers)
I bx ≡ 1(modp) x is an inverse for b!
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 47: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/47.jpg)
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Probability on Finite GroupsMinkowski functionals and Valuations
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Orbit counting Theorem
Aliases:Burnside’s LemmaBurnside’s counting theoremThe theorem that is not Burnside’sThe Cauchy-Frobenious lemma, so on.Number of orbits=average number of point fixed by the action ofelements of G.|X/G | = 1
|G |∑
g∈G |X g |
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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Probability on Finite GroupsMinkowski functionals and Valuations
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Coloring
What is the number R of rotationally distinct colorings of a the facesof a cube using 3 colors?Let the X be the set of all 36 = 729 colored cubes. Two elements arein the same orbit precisely when one is a rotation (or composition ofrotations) of the other.
I 1 identity fix all 36 elements of X
I 6 90 degree face rotations fix 33
I 3 180 degree face rotations fix 34
I 8 120 degree vertex rotation fix 32
I 6 180 edge rotations fix 33
So N = 124(1 ∗ 36 + 6 ∗ 33 + 3 ∗ 34 + 8 ∗ 32 + 6 ∗ 33) = 57
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 49: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/49.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Coloring
What is the number R of rotationally distinct colorings of a the facesof a cube using 3 colors?Let the X be the set of all 36 = 729 colored cubes. Two elements arein the same orbit precisely when one is a rotation (or composition ofrotations) of the other.
I 1 identity fix all 36 elements of XI 6 90 degree face rotations fix 33
I 3 180 degree face rotations fix 34
I 8 120 degree vertex rotation fix 32
I 6 180 edge rotations fix 33
So N = 124(1 ∗ 36 + 6 ∗ 33 + 3 ∗ 34 + 8 ∗ 32 + 6 ∗ 33) = 57
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 50: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/50.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Coloring
What is the number R of rotationally distinct colorings of a the facesof a cube using 3 colors?Let the X be the set of all 36 = 729 colored cubes. Two elements arein the same orbit precisely when one is a rotation (or composition ofrotations) of the other.
I 1 identity fix all 36 elements of XI 6 90 degree face rotations fix 33
I 3 180 degree face rotations fix 34
I 8 120 degree vertex rotation fix 32
I 6 180 edge rotations fix 33
So N = 124(1 ∗ 36 + 6 ∗ 33 + 3 ∗ 34 + 8 ∗ 32 + 6 ∗ 33) = 57
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 51: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/51.jpg)
OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Coloring
What is the number R of rotationally distinct colorings of a the facesof a cube using 3 colors?Let the X be the set of all 36 = 729 colored cubes. Two elements arein the same orbit precisely when one is a rotation (or composition ofrotations) of the other.
I 1 identity fix all 36 elements of XI 6 90 degree face rotations fix 33
I 3 180 degree face rotations fix 34
I 8 120 degree vertex rotation fix 32
I 6 180 edge rotations fix 33
So N = 124(1 ∗ 36 + 6 ∗ 33 + 3 ∗ 34 + 8 ∗ 32 + 6 ∗ 33) = 57
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
![Page 52: Aspects of Group Theory in Stochastic Problemsssll.cecs.anu.edu.au/files/slides/barbosa.pdf · Probability on Finite Groups ... Algebraic and Numeric Programming environments Aspects](https://reader034.vdocuments.net/reader034/viewer/2022043018/5f3af3e9b13d4623b75e38fe/html5/thumbnails/52.jpg)
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Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Finite GroupSubgroups and CosetsLagrange TheoremOrbit CountingRelated issues
Coloring
What is the number R of rotationally distinct colorings of a the facesof a cube using 3 colors?Let the X be the set of all 36 = 729 colored cubes. Two elements arein the same orbit precisely when one is a rotation (or composition ofrotations) of the other.
I 1 identity fix all 36 elements of XI 6 90 degree face rotations fix 33
I 3 180 degree face rotations fix 34
I 8 120 degree vertex rotation fix 32
I 6 180 edge rotations fix 33
So N = 124(1 ∗ 36 + 6 ∗ 33 + 3 ∗ 34 + 8 ∗ 32 + 6 ∗ 33) = 57
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orbit-stabilizer
g1
g2
gn
eGx
Gx = {g ! G|g.x = x}
x
g1
g2x
x
gn
g !" g.xh
x
one orbit
|G/Gx| = |G(x)|
G(x) = {g.x|g ! G}
G/N = {a.N |a ! G}
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orbit-counting proof
Define the set of all fixed points by an element of G byX g = {x ∈ X |gx = x} and the set of all orbits byX/G = {G (x)|x ∈ X}.∑
g∈G
|X g | =∑g∈G
(∑
x :gx=x
1) =∑x∈X
(∑
g :gx=x
1)
=∑x∈X
|Gx | =∑x∈X
|G ||G (x)|
= |G |∑
ω∈X/G
∑x∈ω
(1
ω)
= |G ||X/G | |ω||ω|
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Structure and Symmetry
I The number of ways of coloring the elements of an n-elementset X with k colors is kn.
I A structural refinement is to have a graph Γ on the vertex setX and count proper coloring of Γ. The answer is a polynomialof degree n in k, the chromatic polynomialχ(Γ; k).
I A refinement involving symmetry is to have a group G ofpermutations of X , and to count colorings up to the action ofG . The answer is what we saw before, the orbit countingtheorem: a polynomial of degree n in k, with leadingcoefficient 1/|G |.
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Structure and Symmetry
I The number of ways of coloring the elements of an n-elementset X with k colors is kn.
I A structural refinement is to have a graph Γ on the vertex setX and count proper coloring of Γ. The answer is a polynomialof degree n in k, the chromatic polynomialχ(Γ; k).
I A refinement involving symmetry is to have a group G ofpermutations of X , and to count colorings up to the action ofG . The answer is what we saw before, the orbit countingtheorem: a polynomial of degree n in k, with leadingcoefficient 1/|G |.
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Structure and Symmetry
I The number of ways of coloring the elements of an n-elementset X with k colors is kn.
I A structural refinement is to have a graph Γ on the vertex setX and count proper coloring of Γ. The answer is a polynomialof degree n in k, the chromatic polynomialχ(Γ; k).
I A refinement involving symmetry is to have a group G ofpermutations of X , and to count colorings up to the action ofG . The answer is what we saw before, the orbit countingtheorem: a polynomial of degree n in k, with leadingcoefficient 1/|G |.
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I Combining the two approaches leads to counting the G-orbitsof “structurally restricted” where G is the group ofautomorphisms of the structure imposed on X. The answer isthe orbital chromatic polynomial of (Γ,G ). [P.J. Cameron, B.Jackson and Jason Rudd, 2006.]
I The Tutte polynomial (a generalization of the chromaticpolynomial) is are related to q-state Potts model partitionfunction in the Fortuin-Kastelyn representation. [J. J.Jacobsen, J. Salas, A. D. Sokal, 2005].
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I Combining the two approaches leads to counting the G-orbitsof “structurally restricted” where G is the group ofautomorphisms of the structure imposed on X. The answer isthe orbital chromatic polynomial of (Γ,G ). [P.J. Cameron, B.Jackson and Jason Rudd, 2006.]
I The Tutte polynomial (a generalization of the chromaticpolynomial) is are related to q-state Potts model partitionfunction in the Fortuin-Kastelyn representation. [J. J.Jacobsen, J. Salas, A. D. Sokal, 2005].
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Fortuin-Kastelyn representation
H(σ) = −∑
e=ij∈E
Jeδ(σi , σj)
Z =∑
σ
e−βH(σ)
ZG (q, v) ==∑
σ
∏e=ij∈E
{1 + veδ(σi , σj)}
where
ve = eβJe − 1
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Fortuin-Kastelyn representation
ZG (q, v) = q|V |∑A⊆E
qc(Z)∏e∈A
ve
q
TG (x , y) =∑A⊆E
(x − 1)k(A)−k(G)(y − 1)c(A)
TG (x , y) = (x − 1)−k(G)(y − 1)|V |ZG ((x − 1)(y − 1), y − 1).
Why they do this? In order to study the limit q → 0, singularitiestells about phase change...
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Mallows Models
I Exact results
I Approximations (sampling)
I Generalization to partial rankings (Guy Lebanon, nips2007)
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I Exact results
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I Generalization to partial rankings (Guy Lebanon, nips2007)
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Mallows-Type Models
idea
(1)thought
(2)play
(3)theory
(4)dream
(5)attention
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Distribution on permutations
Distance between permutations, Kendall’s tau:
d(π, σ) =n−1∑i=1
∑l>i
I (πσ−1(i)− πσ−1(l))
d(π, σ) = i(πσ−1) = i(κ)
Equivalent to the number of adjacent transpositions needed tobring π−1 to σ−1.
pσ(π) =1
Z (c)e−cd(π,σ)
Z (c) =∑π∈Gn
e−cd(π,σ)
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Evaluating Z (c)
For q > 0, [ see Stanley, 2000 ]
∑π∈Gn
qi(π) =n−1∑a1=0
n−2∑a2=0
...
0∑an=0
qa1+a2+...+an
= (n−1∑a1=0
qa1)(n−2∑a2=0
qa2)...(0∑
an=0
qan)
= (1 + q + ... + qn−1)...(1 + q + q2)(1 + q)1.
=n−1∏j=1
j∑k=0
qk
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Evaluating Z (c)
Z (c) =∑π∈Gn
e−ci(κ)
= (1 + e−c + ... + e−(n−1)c)...(1 + e−c + e−2c)(1 + e−c)
=n∏
j=1
1− e−jc
1− e−c
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Partial Ranks and Cosets
G1,1,2 = {e, (3, 4)} G1,...1,n!k = {! ! Gn|!(i) = i, i = 1...k}
G1,...,1,n!k! = {"!|" ! G1,..,1,n!k}
G1,1,2!
Set of permutations consistent with the ordering ! on the k top-ranked
One of the cosets of G1,...,1,n!k ! Gn, indexed by ! " Gn
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Fourier transform on groups
P̂(ρ) =∑s∈G
P(s)ρ(s)
f̂ (w) =1√2π
∫ ∞
−∞f (x)e−iwxdx
Upper bound lemma: Let Q be a probability on the finite group G andU the uniform distribution:
|Q − U|2 ≤ 1
4
∑ρ
dρTr(Q̂(ρ)Q̂(ρ)†)
The metric here is the total variation distance, related to other metricssuch as Hellinger distance and Kullback-Leibler separation.
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Fastest mixing Markov Chain with symmetries
I Transitions are interpreted as convolutions.
I Upper bound lemma used to estimate convergence.
I Symmetry (automorphism group) is used to reduce thenumber of variables.
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Fastest mixing Markov Chain with symmetries
I Transitions are interpreted as convolutions.
I Upper bound lemma used to estimate convergence.
I Symmetry (automorphism group) is used to reduce thenumber of variables.
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I Transitions are interpreted as convolutions.
I Upper bound lemma used to estimate convergence.
I Symmetry (automorphism group) is used to reduce thenumber of variables.
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Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Steiner FormulaDecomposition into open bodiesValuations MotivationMinkowski ValuationsMorphological CentroidsPrevious Applications
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Steiner FormulaDecomposition into open bodiesValuations MotivationMinkowski ValuationsMorphological CentroidsPrevious Applications
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Protein Structure ClassificationHarmonic Analysis on SE(3): Spherical Filters
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Protein Structure ClassificationHarmonic Analysis on SE(3): Spherical Filters
F(f ) = f̂ (p) =
∫SE(3)
f (g)U(g−1, p)d(g) (1)
f (g) = F−1(f̂ ) =1
2π2
∫SE(3)
trace(f̂ (p)U(g , p))p2dp (2)
(f1 ∗ f2)(g) =
∫SE(3)
f1(h)f2(h−1og)d(h) (3)
F(f1 ∗ f2) = F2F1 (4)
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Tools
I Gap
I Grape
I Mathematica, Matlab and Maple
I Snob++
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Tools
I Gap
I Grape
I Mathematica, Matlab and Maple
I Snob++
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Tools
I Gap
I Grape
I Mathematica, Matlab and Maple
I Snob++
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Tools
I Gap
I Grape
I Mathematica, Matlab and Maple
I Snob++
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Thanks and take care!
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems
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OutlineGroups
Probability on Finite GroupsMinkowski functionals and Valuations
ApplicationsAlgebraic and Numeric Programming environments
Thanks and take care!
Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia Aspects of Group Theory in Stochastic Problems