Introduction to Applied Algebraic Topology: Persistent Homology

Introduction
Topology
Applying Topology
Examples
Introduction to Applied
Algebraic Topology:
Persistent Homology
Joshua Roberts
Department of Mathematics
University of Kentucky
September 23, 2009
Joshua Roberts Applied Algebraic Topology 1/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 2/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology? (The
Topology Part)
Joshua Roberts Applied Algebraic Topology 3/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology? (The
Topology Part)
• Topology - Study of shapes that can be deformed
continuously into other shapes
Joshua Roberts Applied Algebraic Topology 3/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology? (The
Topology Part)
• Topology - Study of shapes that can be deformed
continuously into other shapes
• The surface of a cube can be “flattened” to a sphere, they
are homeomorphic (topologically equivalent)
Joshua Roberts Applied Algebraic Topology 3/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology? (The
Topology Part)
• Topology - Study of shapes that can be deformed
continuously into other shapes
• The surface of a cube can be “flattened” to a sphere, they
are homeomorphic (topologically equivalent)
...also...
Joshua Roberts Applied Algebraic Topology 3/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology? (The
Topology Part)
• Topology - Study of shapes that can be deformed
continuously into other shapes
• The surface of a cube can be “flattened” to a sphere, they
are homeomorphic (topologically equivalent)
...also...
Click Me!!!
Joshua Roberts Applied Algebraic Topology 3/ 42

Introduction
Topology
Applying Topology
Examples
Homotopy Equivalence
Joshua Roberts Applied Algebraic Topology 4/ 42

Introduction
Topology
Applying Topology
Examples
Homotopy Equivalence
• Problem - Homeomorphism is too strong!
Joshua Roberts Applied Algebraic Topology 4/ 42

Introduction
Topology
Applying Topology
Examples
Homotopy Equivalence
• Problem - Homeomorphism is too strong! Difficult to
classify topological spaces up to homeomorphism
Joshua Roberts Applied Algebraic Topology 4/ 42

Introduction
Topology
Applying Topology
Examples
Homotopy Equivalence
• Problem - Homeomorphism is too strong! Difficult to
classify topological spaces up to homeomorphism
• Solution - Homotopy!
Joshua Roberts Applied Algebraic Topology 4/ 42

Introduction
Topology
Applying Topology
Examples
Homotopy Equivalence
• Problem - Homeomorphism is too strong! Difficult to
classify topological spaces up to homeomorphism
• Solution - Homotopy! Intuitively, two spaces are homotopy
equivalent if one can be deformed into the other, allowing
contractions and expansion
Joshua Roberts Applied Algebraic Topology 4/ 42

Introduction
Topology
Applying Topology
Examples
Homotopy Equivalence
• Problem - Homeomorphism is too strong! Difficult to
classify topological spaces up to homeomorphism
• Solution - Homotopy! Intuitively, two spaces are homotopy
equivalent if one can be deformed into the other, allowing
contractions and expansion
• Ex:
Joshua Roberts Applied Algebraic Topology 4/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology?
Joshua Roberts Applied Algebraic Topology 5/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology?
• Algebraic topology assigns algebraic invariants to
topological spaces.
Joshua Roberts Applied Algebraic Topology 5/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology?
• Algebraic topology assigns algebraic invariants to
topological spaces.
• χ - Euler characteristic
Joshua Roberts Applied Algebraic Topology 5/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology?
• Algebraic topology assigns algebraic invariants to
topological spaces.
• χ - Euler characteristic - Assigns integers to topological
spaces
Joshua Roberts Applied Algebraic Topology 5/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology?
• Algebraic topology assigns algebraic invariants to
topological spaces.
• χ - Euler characteristic - Assigns integers to topological
spaces - Homotopy Invariant
Joshua Roberts Applied Algebraic Topology 5/ 42

Introduction
Topology
Applying Topology
Examples
What is Algebraic Topology?
• Algebraic topology assigns algebraic invariants to
topological spaces.
• χ - Euler characteristic - Assigns integers to topological
spaces - Homotopy Invariant
• If X is homotopy equivalent to a polyhedron,
χ(X) = #v − #e + #f
Joshua Roberts Applied Algebraic Topology 5/ 42

Introduction
Topology
Applying Topology
Examples
Euler characteristic
Theorem
The Euler characteristic completely classifies (compact,
oriented) surfaces.
Joshua Roberts Applied Algebraic Topology 6/ 42

Introduction
Topology
Applying Topology
Examples
Euler characteristic
Theorem
The Euler characteristic completely classifies (compact,
oriented) surfaces.
Joshua Roberts Applied Algebraic Topology 6/ 42

Introduction
Topology
Applying Topology
Examples
Euler characteristic
Theorem
The Euler characteristic completely classifies (compact,
oriented) surfaces.
Joshua Roberts Applied Algebraic Topology 6/ 42

Introduction
Topology
Applying Topology
Examples
Euler characteristic
Theorem
The Euler characteristic completely classifies (compact,
oriented) surfaces.
Joshua Roberts Applied Algebraic Topology 6/ 42

Introduction
Topology
Applying Topology
Examples
Euler characteristic
Theorem
The Euler characteristic completely classifies (compact,
oriented) surfaces.
Joshua Roberts Applied Algebraic Topology 6/ 42

Introduction
Topology
Applying Topology
Examples
Euler characteristic
Theorem
The Euler characteristic completely classifies (compact,
oriented) surfaces.
Joshua Roberts Applied Algebraic Topology 6/ 42

Introduction
Topology
Applying Topology
Examples
“Topology! The stratosphere of human thought! In the
twenty-fourth century it might possibly be of use to someone...”
The First Circle, A. Solzhenitsyn
Joshua Roberts Applied Algebraic Topology 7/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 8/ 42

Simplices
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Joshua Roberts Applied Algebraic Topology 9/ 42

Simplices
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• A simplex is the convex hull of a collection of points in
“general position”
Joshua Roberts Applied Algebraic Topology 9/ 42

Simplices
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• A simplex is the convex hull of a collection of points in
“general position”
• A 4-simplex is ...
Joshua Roberts Applied Algebraic Topology 9/ 42

Simplices
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• A simplex is the convex hull of a collection of points in
“general position”
• A 4-simplex is ... (you get the idea)
Joshua Roberts Applied Algebraic Topology 9/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complexes
Simplicial Complex
Homology
• A simplicial complex is a bunch of simplices glued together
so that the intersection of any two simplicies is another
simplex.
Joshua Roberts Applied Algebraic Topology 10/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complexes
Simplicial Complex
Homology
• A simplicial complex is a bunch of simplices glued together
so that the intersection of any two simplicies is another
simplex.
Figure:
Simplicial
Complexes
Joshua Roberts Applied Algebraic Topology 10/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complexes
Simplicial Complex
Homology
• A simplicial complex is a bunch of simplices glued together
so that the intersection of any two simplicies is another
simplex.
Figure:
Simplicial
Complexes
Figure: Not a
Simplicial
Complex
Joshua Roberts Applied Algebraic Topology 10/ 42

Triangulations
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Theorem
Every topological space (manifold) up to dimension three is
homeomorphic to a simplicial complex.
Joshua Roberts Applied Algebraic Topology 11/ 42

Triangulations
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Theorem
Every topological space (manifold) up to dimension three is
homeomorphic to a simplicial complex.
This is called a triangulation of the space.
Joshua Roberts Applied Algebraic Topology 11/ 42

Triangulations
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Theorem
Every topological space (manifold) up to dimension three is
homeomorphic to a simplicial complex.
This is called a triangulation of the space.
Joshua Roberts Applied Algebraic Topology 11/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Complexes to Vector Spaces
• Given a simplicial complex, we can associate vector
spaces (over a chosen field) corresponding to n-simplicies,
where a simplex is a basis element for the vector space.
Joshua Roberts Applied Algebraic Topology 12/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Complexes to Vector Spaces
• Given a simplicial complex, we can associate vector
spaces (over a chosen field) corresponding to n-simplicies,
where a simplex is a basis element for the vector space.
Let X be the simplicial complex ...
Joshua Roberts Applied Algebraic Topology 12/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Complexes to Vector Spaces
• Given a simplicial complex, we can associate vector
spaces (over a chosen field) corresponding to n-simplicies,
where a simplex is a basis element for the vector space.
Let X be the simplicial complex ...
• C 0 (X) = F 2v × F 2w × F 2x × F 2y × F 2z
Joshua Roberts Applied Algebraic Topology 12/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Complexes to Vector Spaces
• Given a simplicial complex, we can associate vector
spaces (over a chosen field) corresponding to n-simplicies,
where a simplex is a basis element for the vector space.
Let X be the simplicial complex ...
• C 0 (X) = F 2v × F 2w × F 2x × F 2y × F 2z
• C 1 (X) = F 2a × F 2b × F 2c × F 2d × F 2e × F 2f
Joshua Roberts Applied Algebraic Topology 12/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Complexes to Vector Spaces
• Given a simplicial complex, we can associate vector
spaces (over a chosen field) corresponding to n-simplicies,
where a simplex is a basis element for the vector space.
Let X be the simplicial complex ...
• C 0 (X) = F 2v × F 2w × F 2x × F 2y × F 2z
• C 1 (X) = F 2a × F 2b × F 2c × F 2d × F 2e × F 2f
• C 2 (X) = F 2K
Joshua Roberts Applied Algebraic Topology 12/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Complexes to Vector Spaces
• Given a simplicial complex, we can associate vector
spaces (over a chosen field) corresponding to n-simplicies,
where a simplex is a basis element for the vector space.
Let X be the simplicial complex ...
• C 0 (X) = F 2v × F 2w × F 2x × F 2y × F 2z
• C 1 (X) = F 2a × F 2b × F 2c × F 2d × F 2e × F 2f
• C 2 (X) = F 2K
• C n (X) = 0 for n ≥ 3
Joshua Roberts Applied Algebraic Topology 12/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 13/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
• Given a simplicial complex X, Define the
n th homology of X to be the quotient
vector space
Joshua Roberts Applied Algebraic Topology 14/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
• Given a simplicial complex X, Define the
n th homology of X to be the quotient
vector space
H n (X) = “Cycles”/“Boundaries”
Joshua Roberts Applied Algebraic Topology 14/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
• Given a simplicial complex X, Define the
n th homology of X to be the quotient
vector space
H n (X) = “Cycles”/“Boundaries”
H 1 (X)
= ([a + b + c] + [d + e + f ])/∂K
Joshua Roberts Applied Algebraic Topology 14/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
• Given a simplicial complex X, Define the
n th homology of X to be the quotient
vector space
H n (X) = “Cycles”/“Boundaries”
H 1 (X)
= ([a + b + c] + [d + e + f ])/∂K
= ([a + b + c] + [d + e + f ])/[a + b + c]
Joshua Roberts Applied Algebraic Topology 14/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
• Given a simplicial complex X, Define the
n th homology of X to be the quotient
vector space
H n (X) = “Cycles”/“Boundaries”
H 1 (X)
= ([a + b + c] + [d + e + f ])/∂K
= ([a + b + c] + [d + e + f ])/[a + b + c]
∼= [d + e + f ]
Joshua Roberts Applied Algebraic Topology 14/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
• Given a simplicial complex X, Define the
n th homology of X to be the quotient
vector space
H n (X) = “Cycles”/“Boundaries”
H 1 (X)
= ([a + b + c] + [d + e + f ])/∂K
= ([a + b + c] + [d + e + f ])/[a + b + c]
∼= [d + e + f ]
∼= F 2
Joshua Roberts Applied Algebraic Topology 14/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
The intuition is that homology measures “holes”
Joshua Roberts Applied Algebraic Topology 15/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
The intuition is that homology measures “holes”
• 0-dimensional holes are connected components
Joshua Roberts Applied Algebraic Topology 15/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
The intuition is that homology measures “holes”
• 0-dimensional holes are connected components
• 1-dimensional holes are circles
Joshua Roberts Applied Algebraic Topology 15/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
The intuition is that homology measures “holes”
• 0-dimensional holes are connected components
• 1-dimensional holes are circles
• 2-dimensional holes are cavities
Joshua Roberts Applied Algebraic Topology 15/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
The intuition is that homology measures “holes”
• 0-dimensional holes are connected components
• 1-dimensional holes are circles
• 2-dimensional holes are cavities
• 3-dimensional holes are ...
Joshua Roberts Applied Algebraic Topology 15/ 42

Introduction
Topology
Applying Topology
Examples
Simplicial Homology
Simplicial Complex
Homology
The intuition is that homology measures “holes”
• 0-dimensional holes are connected components
• 1-dimensional holes are circles
• 2-dimensional holes are cavities
• 3-dimensional holes are ... (you get the idea)
Joshua Roberts Applied Algebraic Topology 15/ 42

Betti Numbers
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
Joshua Roberts Applied Algebraic Topology 16/ 42

Betti Numbers
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• The n th Betti number of a simplicial complex X is the
dimension of the vector space H n (X).
Joshua Roberts Applied Algebraic Topology 16/ 42

Betti Numbers
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• The n th Betti number of a simplicial complex X is the
dimension of the vector space H n (X).
• This is also the number of n-dimensional holes in the
simplex
Joshua Roberts Applied Algebraic Topology 16/ 42

Betti Numbers
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• The n th Betti number of a simplicial complex X is the
dimension of the vector space H n (X).
• This is also the number of n-dimensional holes in the
simplex
Joshua Roberts Applied Algebraic Topology 16/ 42

Betti Numbers
Introduction
Topology
Applying Topology
Examples
Simplicial Complex
Homology
• The n th Betti number of a simplicial complex X is the
dimension of the vector space H n (X).
• This is also the number of n-dimensional holes in the
simplex
Joshua Roberts Applied Algebraic Topology 16/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 17/ 42

Point-Clouds
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 18/ 42

Point-Clouds
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Let X be a collection of points in Euclidean space R n
Joshua Roberts Applied Algebraic Topology 18/ 42

Point-Clouds
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Let X be a collection of points in Euclidean space R n
• In practice, X is obtained from data coming from
questionnaires, sensor readings, population size, etc
Joshua Roberts Applied Algebraic Topology 18/ 42

Point-Clouds
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Let X be a collection of points in Euclidean space R n
• In practice, X is obtained from data coming from
questionnaires, sensor readings, population size, etc
• Assumption: The data come from a (potentially noisy)
sampling of points from an underlying topological space (or
manifold)
Joshua Roberts Applied Algebraic Topology 18/ 42

Point-Clouds
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Let X be a collection of points in Euclidean space R n
• In practice, X is obtained from data coming from
questionnaires, sensor readings, population size, etc
• Assumption: The data come from a (potentially noisy)
sampling of points from an underlying topological space (or
manifold)
Figure: Planar projection of higher dimensional point cloud data set
Joshua Roberts Applied Algebraic Topology 18/ 42

Introduction
Topology
Applying Topology
Examples
Topological Methods
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 19/ 42

Introduction
Topology
Applying Topology
Examples
Topological Methods
Point-Cloud Data
Persistence
• We analyze a point cloud data set by putting a simplicial
structure on X and calculating H n (X)
Joshua Roberts Applied Algebraic Topology 19/ 42

Introduction
Topology
Applying Topology
Examples
Topological Methods
Point-Cloud Data
Persistence
• We analyze a point cloud data set by putting a simplicial
structure on X and calculating H n (X)
• Čech complex & Rips complex
Joshua Roberts Applied Algebraic Topology 19/ 42

Čech Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 20/ 42

Čech Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, put a ball of radius ɛ/2 around
each point
Joshua Roberts Applied Algebraic Topology 20/ 42

Čech Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, put a ball of radius ɛ/2 around
each point
• The Čech complex, C ɛ , is the simplicial complex whose
k-simplicies are k-fold intersections of the ɛ/2-balls
Joshua Roberts Applied Algebraic Topology 20/ 42

Čech Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, put a ball of radius ɛ/2 around
each point
• The Čech complex, C ɛ , is the simplicial complex whose
k-simplicies are k-fold intersections of the ɛ/2-balls
Joshua Roberts Applied Algebraic Topology 20/ 42

Čech Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, put a ball of radius ɛ/2 around
each point
• The Čech complex, C ɛ , is the simplicial complex whose
k-simplicies are k-fold intersections of the ɛ/2-balls
Theorem (Nerve Lemma)
The Čech complex has the homotopy type of the cover by
ɛ/2-balls.
Joshua Roberts Applied Algebraic Topology 20/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 21/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, the Rips complex R ɛ is the
simplicial complex formed by:
Joshua Roberts Applied Algebraic Topology 21/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, the Rips complex R ɛ is the
simplicial complex formed by:
• Each set of k points within ɛ of each other forms a
(k − 1)-simplex
Joshua Roberts Applied Algebraic Topology 21/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, the Rips complex R ɛ is the
simplicial complex formed by:
• Each set of k points within ɛ of each other forms a
(k − 1)-simplex
• Two points within ɛ - edge
Joshua Roberts Applied Algebraic Topology 21/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, the Rips complex R ɛ is the
simplicial complex formed by:
• Each set of k points within ɛ of each other forms a
(k − 1)-simplex
• Two points within ɛ - edge
• Three points within ɛ - triangle
Joshua Roberts Applied Algebraic Topology 21/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, the Rips complex R ɛ is the
simplicial complex formed by:
• Each set of k points within ɛ of each other forms a
(k − 1)-simplex
• Two points within ɛ - edge
• Three points within ɛ - triangle
• Four points - tetrahedron
Joshua Roberts Applied Algebraic Topology 21/ 42

Rips Complex
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given a set of points X, the Rips complex R ɛ is the
simplicial complex formed by:
• Each set of k points within ɛ of each other forms a
(k − 1)-simplex
• Two points within ɛ - edge
• Three points within ɛ - triangle
• Four points - tetrahedron
• Etc...
Joshua Roberts Applied Algebraic Topology 21/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 22/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
C ɛ ≃ S 1 ∨ S 1 ∨ S 1 and R ɛ ≃ S 1 ∨ S 2
Joshua Roberts Applied Algebraic Topology 22/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good:
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad:
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Rips Complex
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Rips Complex
• Good:
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Rips Complex
• Good: Combinatorially simpler ⇒ Computationally
cheaper
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Rips Complex
• Good: Combinatorially simpler ⇒ Computationally
cheaper
• Bad:
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Rips Complex
• Good: Combinatorially simpler ⇒ Computationally
cheaper
• Bad: Not homotopy equivalent to union of balls
Joshua Roberts Applied Algebraic Topology 23/ 42

Čech vs. Rips
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Čech Complex
• Good: Homotopy equivalent to the union of balls
• Bad: Can grow very big ⇒ Computationally expensive
Rips Complex
• Good: Combinatorially simpler ⇒ Computationally
cheaper
• Bad: Not homotopy equivalent to union of balls
To compensate we use (a version of) the Rips complex, but
vary ɛ
Joshua Roberts Applied Algebraic Topology 23/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given point cloud data obtained from an annulus, we have
a sequence of Rips complexes by increasing ɛ.
Joshua Roberts Applied Algebraic Topology 24/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• Given point cloud data obtained from an annulus, we have
a sequence of Rips complexes by increasing ɛ.
• The fundamental question is “Which holes are real
topological features?”
Joshua Roberts Applied Algebraic Topology 24/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 25/ 42

Introduction
Topology
Applying Topology
Examples
Persistent Homology
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 26/ 42

Introduction
Topology
Applying Topology
Examples
Persistent Homology
Point-Cloud Data
Persistence
• We formalize and attempt to answer with the concept of
persistent homology
Joshua Roberts Applied Algebraic Topology 26/ 42

Introduction
Topology
Applying Topology
Examples
Persistent Homology
Point-Cloud Data
Persistence
• We formalize and attempt to answer with the concept of
persistent homology
• Given a simplicial complex X, a filtration F of X is a
sequence of subcomplexes such that
Joshua Roberts Applied Algebraic Topology 26/ 42

Introduction
Topology
Applying Topology
Examples
Persistent Homology
Point-Cloud Data
Persistence
• We formalize and attempt to answer with the concept of
persistent homology
• Given a simplicial complex X, a filtration F of X is a
sequence of subcomplexes such that
∅ = X 0 ⊂ X 1 ⊂ · · · ⊂ X n−1 ⊂ X n = X
Joshua Roberts Applied Algebraic Topology 26/ 42

Introduction
Topology
Applying Topology
Examples
For the simplicial complex
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 27/ 42

Introduction
Topology
Applying Topology
Examples
For the simplicial complex
Point-Cloud Data
Persistence
A filtration is
Joshua Roberts Applied Algebraic Topology 27/ 42

Introduction
Topology
Applying Topology
Examples
For the simplicial complex
Point-Cloud Data
Persistence
A filtration is
Joshua Roberts Applied Algebraic Topology 27/ 42

Introduction
Topology
Applying Topology
Examples
For the simplicial complex
Point-Cloud Data
Persistence
A filtration is
• We form the Rips complex at each stage 0 ≤ i ≤ n and
look at the homology
Joshua Roberts Applied Algebraic Topology 27/ 42

Introduction
Topology
Applying Topology
Examples
For the simplicial complex
Point-Cloud Data
Persistence
A filtration is
• We form the Rips complex at each stage 0 ≤ i ≤ n and
look at the homology
• Topological features that persist are considered significant
Joshua Roberts Applied Algebraic Topology 27/ 42

Barcodes
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
We record the homology by barcodes displaying the Betti
numbers for levels of our filtration
Joshua Roberts Applied Algebraic Topology 28/ 42

Barcodes
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
We record the homology by barcodes displaying the Betti
numbers for levels of our filtration
Joshua Roberts Applied Algebraic Topology 28/ 42

Barcodes
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
We record the homology by barcodes displaying the Betti
numbers for levels of our filtration
Joshua Roberts Applied Algebraic Topology 28/ 42

Barcodes
Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
We record the homology by barcodes displaying the Betti
numbers for levels of our filtration
Joshua Roberts Applied Algebraic Topology 28/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 29/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Suppose a data set X yields the following barcodes
Joshua Roberts Applied Algebraic Topology 29/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Suppose a data set X yields the following barcodes
Joshua Roberts Applied Algebraic Topology 29/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Suppose a data set X yields the following barcodes
Joshua Roberts Applied Algebraic Topology 29/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
• One persistent Betti number in dimension 0
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
• One persistent Betti number in dimension 0
• Two persistent Betti numbers in dimension 1
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
• One persistent Betti number in dimension 0
• Two persistent Betti numbers in dimension 1
• No homology in dimension > 1
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
• One persistent Betti number in dimension 0
• Two persistent Betti numbers in dimension 1
• No homology in dimension > 1
• What is the space?
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
• One persistent Betti number in dimension 0
• Two persistent Betti numbers in dimension 1
• No homology in dimension > 1
• What is the space?
Joshua Roberts Applied Algebraic Topology 30/ 42

Introduction
Topology
Applying Topology
Examples
Point-Cloud Data
Persistence
• LOTS of noise in dimension 0 at the beginning
• Noise in dimension 1
• One persistent Betti number in dimension 0
• Two persistent Betti numbers in dimension 1
• No homology in dimension > 1
• What is the space?
Joshua Roberts Applied Algebraic Topology 30/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 31/ 42

Introduction
Topology
Applying Topology
Examples
Kevin Knudson & Kenneth Brown
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 32/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Kevin Knudson & Kenneth Brown
• Collection of speech signals from 3 individuals (cm, if, mc)
Joshua Roberts Applied Algebraic Topology 32/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Kevin Knudson & Kenneth Brown
• Collection of speech signals from 3 individuals (cm, if, mc)
• Eight different sounds - aa, ae, eh, m, n, f, sh ,z
Joshua Roberts Applied Algebraic Topology 32/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Kevin Knudson & Kenneth Brown
• Collection of speech signals from 3 individuals (cm, if, mc)
• Eight different sounds - aa, ae, eh, m, n, f, sh ,z
• For a given signal, let f (t) be the amplitude of the wave at
time t
Joshua Roberts Applied Algebraic Topology 32/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Kevin Knudson & Kenneth Brown
• Collection of speech signals from 3 individuals (cm, if, mc)
• Eight different sounds - aa, ae, eh, m, n, f, sh ,z
• For a given signal, let f (t) be the amplitude of the wave at
time t
• Form a Betti 0 barcode by “sweeping”
Joshua Roberts Applied Algebraic Topology 32/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Kevin Knudson & Kenneth Brown
• Collection of speech signals from 3 individuals (cm, if, mc)
• Eight different sounds - aa, ae, eh, m, n, f, sh ,z
• For a given signal, let f (t) be the amplitude of the wave at
time t
• Form a Betti 0 barcode by “sweeping”
Joshua Roberts Applied Algebraic Topology 32/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 33/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• There is a metric on barcodes that measures (dis)similarity
Joshua Roberts Applied Algebraic Topology 33/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• There is a metric on barcodes that measures (dis)similarity
• The metric space this forms is called the barcode space
Joshua Roberts Applied Algebraic Topology 33/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• There is a metric on barcodes that measures (dis)similarity
• The metric space this forms is called the barcode space
• Each barcode corresponds to a point in the barcode space
Joshua Roberts Applied Algebraic Topology 33/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• There is a metric on barcodes that measures (dis)similarity
• The metric space this forms is called the barcode space
• Each barcode corresponds to a point in the barcode space
• For each speaker, this gave a point cloud in the barcode
space consisting of 800 points
Joshua Roberts Applied Algebraic Topology 33/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• There is a metric on barcodes that measures (dis)similarity
• The metric space this forms is called the barcode space
• Each barcode corresponds to a point in the barcode space
• For each speaker, this gave a point cloud in the barcode
space consisting of 800 points
• They studied the structure of this point cloud, hoping there
would be eight components, one for each sound
Joshua Roberts Applied Algebraic Topology 33/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 34/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 34/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 34/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Eight components for speakers cm and mc
Joshua Roberts Applied Algebraic Topology 34/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Eight components for speakers cm and mc
• But for speaker if only obtain 3 components
Joshua Roberts Applied Algebraic Topology 34/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Eight components for speakers cm and mc
• But for speaker if only obtain 3 components
• Authors attribute this to noise (literally) in the data
Joshua Roberts Applied Algebraic Topology 34/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Eight components for speakers cm and mc
• But for speaker if only obtain 3 components
• Authors attribute this to noise (literally) in the data
• Further research may lead to speech recognizition
software or even speaker recognition software
Joshua Roberts Applied Algebraic Topology 34/ 42

Outline
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Introduction
Topology
Simplicial Complex
Homology
Applying Topology
Point-Cloud Data
Persistence
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 35/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Peter Bubenik, Peter Kim, et al.
• Measured the cortical thickness of 16 autistic, 12 control
right-handed, age-matched males
Joshua Roberts Applied Algebraic Topology 36/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Peter Bubenik, Peter Kim, et al.
• Measured the cortical thickness of 16 autistic, 12 control
right-handed, age-matched males
Joshua Roberts Applied Algebraic Topology 36/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 37/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 38/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
Joshua Roberts Applied Algebraic Topology 39/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Find the persistent homology of the thickness function
Joshua Roberts Applied Algebraic Topology 40/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Find the persistent homology of the thickness function
• (Generalize the “sweeping” function above)
Joshua Roberts Applied Algebraic Topology 40/ 42

Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Find the persistent homology of the thickness function
• (Generalize the “sweeping” function above)
Joshua Roberts Applied Algebraic Topology 40/ 42

Conclusion
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• It’s been common in the history of mathematics for
“useless” ideas to have applications after all (matrix theory,
group theory, etc)
Joshua Roberts Applied Algebraic Topology 41/ 42

Conclusion
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• It’s been common in the history of mathematics for
“useless” ideas to have applications after all (matrix theory,
group theory, etc)
• Algebraic topology is being applied to sensor networks,
statistics & data aggregation, robot motion, dynamics &
numerical PDE’s, and other areas
Joshua Roberts Applied Algebraic Topology 41/ 42

Conclusion
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• It’s been common in the history of mathematics for
“useless” ideas to have applications after all (matrix theory,
group theory, etc)
• Algebraic topology is being applied to sensor networks,
statistics & data aggregation, robot motion, dynamics &
numerical PDE’s, and other areas
• Applied algebraic topology is in its infancy - lots of
problems (and grants!) are open (and available)
Joshua Roberts Applied Algebraic Topology 41/ 42

Conclusion
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• It’s been common in the history of mathematics for
“useless” ideas to have applications after all (matrix theory,
group theory, etc)
• Algebraic topology is being applied to sensor networks,
statistics & data aggregation, robot motion, dynamics &
numerical PDE’s, and other areas
• Applied algebraic topology is in its infancy - lots of
problems (and grants!) are open (and available)
• Problem: Find better (more accurate) ways to build
simplicial complexes
Joshua Roberts Applied Algebraic Topology 41/ 42

Conclusion
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• It’s been common in the history of mathematics for
“useless” ideas to have applications after all (matrix theory,
group theory, etc)
• Algebraic topology is being applied to sensor networks,
statistics & data aggregation, robot motion, dynamics &
numerical PDE’s, and other areas
• Applied algebraic topology is in its infancy - lots of
problems (and grants!) are open (and available)
• Problem: Find better (more accurate) ways to build
simplicial complexes
• Problem: Statistical analysis
Joshua Roberts Applied Algebraic Topology 41/ 42

Thanks!
Introduction
Topology
Applying Topology
Examples
Phonemes
Autism
• Thank you for listening!
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KENTUCKY
URL: http://www.ms.uky.edu/∼jroberts
E-mail address: jroberts@ms.uky.edu
Joshua Roberts Applied Algebraic Topology 42/ 42