Aspects of Group Theory in Stochastic Problems Dr. Marconi Barbosa - - PowerPoint PPT Presentation

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Aspects of Group Theory in Stochastic Problems

• Dr. Marconi Barbosa

NICTA/ANU, Canberra, Australia November 18, 2008

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Mallow’s Model ◮ Harmonic analysis on manifolds ◮ Fourier transforms on groups ◮ Graph matching: edge info added into node features ◮ What we do know about Metropolis algorithm? Exact

Results?

◮ Fastest Mixing Markov Chains ◮ Parallel Coset enumeration

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

Outline of what is (would be nice) to come...

◮ Lagrange Theorem ◮ Example: Fermat Little theorem and cryptography ◮ Orbit Counting Theorem ◮ Example: Cube orbits ◮ Magic cube group: Scary ◮ More scary: Baby Monster ◮ Freaking out: The Monster group ◮ Group classification: one slide soft crash course ◮ Group representation: one slide hard crash course ◮ Invariance, equivalence and symmetry ◮ Differential invariants, variational problems with symmetry. ◮ Geometric probability, Minkowski functionals and continuous

Groups.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Verducci & Fligner: Distance Based Ranking Models ◮ Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain with

symmetry, 2006.

◮ Fagin: Comparing partial ranks, how efficient? ◮ Guy Lebannon: Partial Rankings and Cosets/Posets

(nips2007)

◮ Risi Kondor:

Multi-Object tracking...with...simmetric group (nips2006) Diffusion Kernel in graphs and other structures (manifold methods)

◮ J.Huang C. Guestrin & L. Gibbas: Polytope projection after

Fourier inverse transform and estimation in Fourier Domain. (nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Verducci & Fligner: Distance Based Ranking Models ◮ Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain with

symmetry, 2006.

◮ Fagin: Comparing partial ranks, how efficient? ◮ Guy Lebannon: Partial Rankings and Cosets/Posets

(nips2007)

◮ Risi Kondor:

Multi-Object tracking...with...simmetric group (nips2006) Diffusion Kernel in graphs and other structures (manifold methods)

◮ J.Huang C. Guestrin & L. Gibbas: Polytope projection after

Fourier inverse transform and estimation in Fourier Domain. (nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Verducci & Fligner: Distance Based Ranking Models ◮ Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain with

symmetry, 2006.

◮ Fagin: Comparing partial ranks, how efficient? ◮ Guy Lebannon: Partial Rankings and Cosets/Posets

(nips2007)

◮ Risi Kondor:

Multi-Object tracking...with...simmetric group (nips2006) Diffusion Kernel in graphs and other structures (manifold methods)

◮ J.Huang C. Guestrin & L. Gibbas: Polytope projection after

Fourier inverse transform and estimation in Fourier Domain. (nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Verducci & Fligner: Distance Based Ranking Models ◮ Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain with

symmetry, 2006.

◮ Fagin: Comparing partial ranks, how efficient? ◮ Guy Lebannon: Partial Rankings and Cosets/Posets

(nips2007)

◮ Risi Kondor:

Multi-Object tracking...with...simmetric group (nips2006) Diffusion Kernel in graphs and other structures (manifold methods)

◮ J.Huang C. Guestrin & L. Gibbas: Polytope projection after

Fourier inverse transform and estimation in Fourier Domain. (nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Verducci & Fligner: Distance Based Ranking Models ◮ Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain with

symmetry, 2006.

◮ Fagin: Comparing partial ranks, how efficient? ◮ Guy Lebannon: Partial Rankings and Cosets/Posets

(nips2007)

◮ Risi Kondor:

Multi-Object tracking...with...simmetric group (nips2006) Diffusion Kernel in graphs and other structures (manifold methods)

◮ J.Huang C. Guestrin & L. Gibbas: Polytope projection after

Fourier inverse transform and estimation in Fourier Domain. (nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Scene Finite Groups crash course outline Motivational Papers

◮ Verducci & Fligner: Distance Based Ranking Models ◮ Diaconis, Boyd & Xiao: Fastest Mixing Markov Chain with

symmetry, 2006.

◮ Fagin: Comparing partial ranks, how efficient? ◮ Guy Lebannon: Partial Rankings and Cosets/Posets

(nips2007)

◮ Risi Kondor:

Multi-Object tracking...with...simmetric group (nips2006) Diffusion Kernel in graphs and other structures (manifold methods)

◮ J.Huang C. Guestrin & L. Gibbas: Polytope projection after

Fourier inverse transform and estimation in Fourier Domain. (nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

A group G is a set S and a binary operation ∗ satisfying two properties:

◮ A) Closure. For a, b ∈ G, then a ∗ b ∈ G. ◮ B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) are

the same. There must be two very special members too:

◮ C) The identity element e is such that:

a ∗ e = e ∗ a = a

◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

A group G is a set S and a binary operation ∗ satisfying two properties:

◮ A) Closure. For a, b ∈ G, then a ∗ b ∈ G. ◮ B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) are

the same. There must be two very special members too:

◮ C) The identity element e is such that:

a ∗ e = e ∗ a = a

◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

A group G is a set S and a binary operation ∗ satisfying two properties:

◮ A) Closure. For a, b ∈ G, then a ∗ b ∈ G. ◮ B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) are

the same. There must be two very special members too:

◮ C) The identity element e is such that:

a ∗ e = e ∗ a = a

◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

A group G is a set S and a binary operation ∗ satisfying two properties:

◮ A) Closure. For a, b ∈ G, then a ∗ b ∈ G. ◮ B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) are

the same. There must be two very special members too:

◮ C) The identity element e is such that:

a ∗ e = e ∗ a = a

◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

A group G is a set S and a binary operation ∗ satisfying two properties:

◮ A) Closure. For a, b ∈ G, then a ∗ b ∈ G. ◮ B) Associativity. The elements (a ∗ b) ∗ c and a ∗ (b ∗ c) are

the same. There must be two very special members too:

◮ C) The identity element e is such that:

a ∗ e = e ∗ a = a

◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

◮ A subgroup of G, is any subset of elements that still forms a

group.

◮ Consider a subgroup H of G. Associated with H we create

the following set gH = {g.h : ∀h ∈ H}, This is one of the left cosets of H in G, 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

◮ A subgroup of G, is any subset of elements that still forms a

group.

◮ Consider a subgroup H of G. Associated with H we create

the following set gH = {g.h : ∀h ∈ H}, This is one of the left cosets of H in G, 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Coset example from naturals

Group: G = Z4 Set S: {0, 1, 2, 3} Operation ∗: integer addition mod 4 Subgroup: H = {0, 2} 0 ∗ H = {0, 2} = H, a trivial coset. 1 ∗ H = {1, 3}, first one here 2 ∗ H = {2, 4} = {2, 0} = H, trivial again 3 ∗ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

The order of every subgroup H divides the order of G.

◮ Show that all cosets of H have the same order. Define

f : aH ⇒ bH by f = ba−1. This is a bijective map with inverse f −1 = ab−1.

◮ Show that cosets from H form a partition of G: Cosets are either

identical or disjoint: every element belongs to only one coset.

◮ Then, the number of elements in G is equal the number of cosets

(index) times the number of elements in each coset (which in turn 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

The order of every subgroup H divides the order of G.

◮ Show that all cosets of H have the same order. Define

f : aH ⇒ bH by f = ba−1. This is a bijective map with inverse f −1 = ab−1.

◮ Show that cosets from H form a partition of G: Cosets are either

identical or disjoint: every element belongs to only one coset.

◮ Then, the number of elements in G is equal the number of cosets

(index) times the number of elements in each coset (which in turn 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

The order of every subgroup H divides the order of G.

◮ Show that all cosets of H have the same order. Define

f : aH ⇒ bH by f = ba−1. This is a bijective map with inverse f −1 = ab−1.

◮ Show that cosets from H form a partition of G: Cosets are either

identical or disjoint: every element belongs to only one coset.

◮ Then, the number of elements in G is equal the number of cosets

(index) times the number of elements in each coset (which in turn 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

The order of every subgroup H divides the order of G.

◮ Show that all cosets of H have the same order. Define

f : aH ⇒ bH by f = ba−1. This is a bijective map with inverse f −1 = ab−1.

◮ Show that cosets from H form a partition of G: Cosets are either

identical or disjoint: every element belongs to only one coset.

◮ Then, the number of elements in G is equal the number of cosets

(index) times the number of elements in each coset (which in turn 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Consequences

The order of any element a of a finite group (i.e. the smallest k for which ak = e divides G) divides the order of G. Because the order of a is the order of a cyclic subgroup generated by a.

◮ Fermat’s Little theorem

ap ≡ a (mod p) ;p prime, a integer

◮ Euler’s Theorem

aφ(n) ≡ 1 (mod n); a co-prime n φ(n)(Euler function) counts the number of co-primes from 1 to n.

◮ Carmichael’s Theorem aλ(n) ≡ 1 (mod n)

λ(n)(Carmichael’s function) gives the smallest integer m for which a(m) ≡ 1 (mod n)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Consequences

The order of any element a of a finite group (i.e. the smallest k for which ak = e divides G) divides the order of G. Because the order of a is the order of a cyclic subgroup generated by a.

◮ Fermat’s Little theorem

ap ≡ a (mod p) ;p prime, a integer

◮ Euler’s Theorem

aφ(n) ≡ 1 (mod n); a co-prime n φ(n)(Euler function) counts the number of co-primes from 1 to n.

◮ Carmichael’s Theorem aλ(n) ≡ 1 (mod n)

λ(n)(Carmichael’s function) gives the smallest integer m for which a(m) ≡ 1 (mod n)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Consequences

The order of any element a of a finite group (i.e. the smallest k for which ak = e divides G) divides the order of G. Because the order of a is the order of a cyclic subgroup generated by a.

◮ Fermat’s Little theorem

ap ≡ a (mod p) ;p prime, a integer

◮ Euler’s Theorem

aφ(n) ≡ 1 (mod n); a co-prime n φ(n)(Euler function) counts the number of co-primes from 1 to n.

◮ Carmichael’s Theorem aλ(n) ≡ 1 (mod n)

λ(n)(Carmichael’s function) gives the smallest integer m for which a(m) ≡ 1 (mod n)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Fermat Little theorem proof by group theory

◮ Basic idea is to recognize G = {1, 2, ..., p − 1}, with the

• peration of multiplication mod p as a Group. Some work to

prove that every element is invertible

◮ Assume that a is an element of G and let k be its order. i.e. ◮ ak ≡ 1 (mod p) ◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Fermat Little theorem proof by group theory

◮ Basic idea is to recognize G = {1, 2, ..., p − 1}, with the

• peration of multiplication mod p as a Group. Some work to

prove that every element is invertible

◮ Assume that a is an element of G and let k be its order. i.e. ◮ ak ≡ 1 (mod p) ◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Fermat Little theorem proof by group theory

◮ Basic idea is to recognize G = {1, 2, ..., p − 1}, with the

• peration of multiplication mod p as a Group. Some work to

prove that every element is invertible

◮ Assume that a is an element of G and let k be its order. i.e. ◮ ak ≡ 1 (mod p) ◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Fermat Little theorem proof by group theory

◮ Basic idea is to recognize G = {1, 2, ..., p − 1}, with the

• peration of multiplication mod p as a Group. Some work to

prove that every element is invertible

◮ Assume that a is an element of G and let k be its order. i.e. ◮ ak ≡ 1 (mod p) ◮ 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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Fermat Little theorem proof by group theory

◮ Invertibility property. ◮ Assume b is co-prime (relative prime) to p. Using Bezout

identity (a linear Diophantine equation)

◮ bx + py = 1 ( x, y integers) ◮ bx ≡ 1(modp) x is an inverse for b!

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Orbit counting Theorem

Aliases: Burnside’s Lemma Burnside’s counting theorem The theorem that is not Burnside’s The Cauchy-Frobenious lemma, so on. Number of orbits=average number of point fixed by the action of elements of G. |X/G| =

1 |G|

• g∈G |X g|
• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Coloring

What is the number R of rotationally distinct colorings of a the faces

• f a cube using 3 colors?

Let the X be the set of all 36 = 729 colored cubes. Two elements are in the same orbit precisely when one is a rotation (or composition of rotations) of the other.

◮ 1 identity fix all 36 elements of X

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Coloring

What is the number R of rotationally distinct colorings of a the faces

• f a cube using 3 colors?

Let the X be the set of all 36 = 729 colored cubes. Two elements are in the same orbit precisely when one is a rotation (or composition of rotations) of the other.

◮ 1 identity fix all 36 elements of X ◮ 6 90 degree face rotations fix 33

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Coloring

What is the number R of rotationally distinct colorings of a the faces

• f a cube using 3 colors?

Let the X be the set of all 36 = 729 colored cubes. Two elements are in the same orbit precisely when one is a rotation (or composition of rotations) of the other.

◮ 1 identity fix all 36 elements of X ◮ 6 90 degree face rotations fix 33 ◮ 3 180 degree face rotations fix 34

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Coloring

What is the number R of rotationally distinct colorings of a the faces

• f a cube using 3 colors?

Let the X be the set of all 36 = 729 colored cubes. Two elements are in the same orbit precisely when one is a rotation (or composition of rotations) of the other.

◮ 1 identity fix all 36 elements of X ◮ 6 90 degree face rotations fix 33 ◮ 3 180 degree face rotations fix 34 ◮ 8 120 degree vertex rotation fix 32

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Coloring

What is the number R of rotationally distinct colorings of a the faces

• f a cube using 3 colors?

Let the X be the set of all 36 = 729 colored cubes. Two elements are in the same orbit precisely when one is a rotation (or composition of rotations) of the other.

◮ 1 identity fix all 36 elements of X ◮ 6 90 degree face rotations fix 33 ◮ 3 180 degree face rotations fix 34 ◮ 8 120 degree vertex rotation fix 32 ◮ 6 180 edge rotations fix 33

So N = 1

24(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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

• rbit-stabilizer

g1 g2 gn e Gx Gx = {g ∈ G|g.x = x} x g1 g2 x x gn g → g.x h x

• ne orbit

|G/Gx| = |G(x)| G(x) = {g.x|g ∈ G}

G/N = {a.N|a ∈ G}

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

• rbit-counting proof

Define the set of all fixed points by an element of G by X g = {x ∈ X|gx = x} and the set of all orbits by X/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|

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Structure and Symmetry

◮ The number of ways of coloring the elements of an n-element

set X with k colors is kn.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Structure and Symmetry

◮ The number of ways of coloring the elements of an n-element

set X with k colors is kn.

◮ A structural refinement is to have a graph Γ on the vertex set

X and count proper coloring of Γ. The answer is a polynomial

• f degree n in k, the chromatic polynomialχ(Γ; k).
• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Structure and Symmetry

◮ The number of ways of coloring the elements of an n-element

set X with k colors is kn.

◮ A structural refinement is to have a graph Γ on the vertex set

X and count proper coloring of Γ. The answer is a polynomial

• f degree n in k, the chromatic polynomialχ(Γ; k).

◮ A refinement involving symmetry is to have a group G of

permutations of X, and to count colorings up to the action of

• G. The answer is what we saw before, the orbit counting

theorem: a polynomial of degree n in k, with leading coefficient 1/|G|.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Structure and Symmetry

◮ Combining the two approaches leads to counting the G-orbits

• f “structurally restricted” where G is the group of

automorphisms of the structure imposed on X. The answer is the orbital chromatic polynomial of (Γ, G). [P.J. Cameron, B. Jackson and Jason Rudd, 2006.]

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

Structure and Symmetry

◮ Combining the two approaches leads to counting the G-orbits

• f “structurally restricted” where G is the group of

automorphisms of the structure imposed on X. The answer is the orbital chromatic polynomial of (Γ, G). [P.J. Cameron, B. Jackson and Jason Rudd, 2006.]

◮ The Tutte polynomial (a generalization of the chromatic

polynomial) is are related to q-state Potts model partition function in the Fortuin-Kastelyn representation. [J. J. Jacobsen, J. Salas, A. D. Sokal, 2005].

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

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

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Finite Group Subgroups and Cosets Lagrange Theorem Orbit Counting Related issues

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, singularities tells about phase change...

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Mallows Models

◮ Exact results

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Mallows Models

◮ Exact results ◮ Approximations (sampling)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Mallows Models

◮ Exact results ◮ Approximations (sampling) ◮ Generalization to partial rankings (Guy Lebanon, nips2007)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Mallows-Type Models

idea (1)thought (2)play (3)theory (4)dream (5)attention

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

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 to bring π−1 to σ−1. pσ(π) = 1 Z(c)e−cd(π,σ) Z(c) =

• π∈Gn

e−cd(π,σ)

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Evaluating Z(c)

For q > 0, [ see Stanley, 2000 ]

• π∈Gn

qi(π) =

n−1

• a1=0

n−2

• a2=0

...

• an=0

qa1+a2+...+an = (

n−1

• a1=0

qa1)(

n−2

• a2=0

qa2)...(

• an=0

qan) = (1 + q + ... + qn−1)...(1 + q + q2)(1 + q)1. =

n−1

• j=1

j

• k=0

qk

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

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

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Partial Ranks and Cosets

G1,1,2 = {e, (3, 4)} G1,...1,n−k = {π ∈ Gn|π(i) = i, i = 1...k} G1,...,1,n−kπ = {σπ|σ ∈ G1,..,1,n−k}

G1,1,2π

Set of permutations consistent with the ordering π on the k top-ranked One of the cosets of G1,...,1,n−k ⊂ Gn, indexed by π ∈ Gn

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Fourier transform on groups

ˆ P(ρ) =

• s∈G

P(s)ρ(s) ˆ f (w) = 1 √ 2π ∞

−∞

f (x)e−iwxdx Upper bound lemma: Let Q be a probability on the finite group G and U the uniform distribution: |Q − U|2 ≤ 1 4

• ρ

dρTr( ˆ Q(ρ) ˆ Q(ρ)†) The metric here is the total variation distance, related to other metrics such as Hellinger distance and Kullback-Leibler separation.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Fastest mixing Markov Chain with symmetries

◮ Transitions are interpreted as convolutions.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Fastest mixing Markov Chain with symmetries

◮ Transitions are interpreted as convolutions. ◮ Upper bound lemma used to estimate convergence.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Fastest mixing Markov Chain with symmetries

◮ Transitions are interpreted as convolutions. ◮ Upper bound lemma used to estimate convergence. ◮ Symmetry (automorphism group) is used to reduce the

number of variables.

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Representations Teaser

Fastest mixing Markov Chain with symmetries

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Steiner Formula Decomposition into open bodies Valuations Motivation Minkowski Valuations Morphological Centroids Previous Applications

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Protein Structure Classification Harmonic Analysis on SE(3): Spherical Filters

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments Protein Structure Classification Harmonic Analysis on SE(3): Spherical Filters

F(f ) = ˆ f (p) =

• SE(3)

f (g)U(g−1, p)d(g) (1) f (g) = F−1(ˆ f ) = 1 2π2

• SE(3)

trace(ˆ f (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

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Tools

◮ Gap

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Tools

◮ Gap ◮ Grape

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Tools

◮ Gap ◮ Grape ◮ Mathematica, Matlab and Maple

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Tools

◮ Gap ◮ Grape ◮ Mathematica, Matlab and Maple ◮ Snob++

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Thanks and take care!

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems

Outline Groups Probability on Finite Groups Minkowski functionals and Valuations Applications Algebraic and Numeric Programming environments

Thanks and take care!

• Dr. Marconi Barbosa NICTA/ANU, Canberra, Australia

Aspects of Group Theory in Stochastic Problems