DDGSpring2016

 

Reading3

《BARCODES: THE PERSISTENT TOPOLOGY OF DATA》比较通俗

《THREE EXAMPLES OF APPLIED & COMPUTATIONAL HOMOLOGY 》 Robert Ghrist，非常通俗

《Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing》很多概念（包括群论）介绍的不错，比较通俗

Definition 4.
Given a subgroup H of an abelian group G, for any a∈G, the set a∗H = {a∗h | h∈H} is the coset of H represented by a.




Definition 7. 
The kernel of a homomorphism φ:G→G' is kerφ = {a∈G | φ(a)=e'}. 
In other words, the kernel is the elements that map to identity.


Define a new binary operation ☆ not on the elements of G but on the cosets of H:
(a∗H) ☆ (b∗H) = (a∗b)∗H, ∀a,b∈G.


Definition 8. 
The cosets {a∗H |a∈G} under the operation ☆ form a group, called the quotient group G/H.

quotient group G/H： <{a∗H |a∈G},☆>

- coset是一个set，(set里的元素是G里的元素)
- quotient group G/H里的每一个元素是(H的)一个coset



Persistent homology finds “holes” by identifying equivalent cycles: Consider the following space in yellow with a small white hole. Imagine the blue cycle as a rubber band. It can be stretched and bent within the space into the green cycle, but not the red one without tearing itself. 
There are two equivalent classes of rubber bands: some surround the hole and others do not. Conversely, two equivalent classes indicate one hole. 

Because kerφ is a subgroup, we can partition G into cosets of the form a∗kerφ for a∈G. These cosets are the white or blue squares.
It is useful to think of quotient groups as “higher level” groups defined on the squares in the previous picture. kerφ (the blue square) is a subgroup of G. The elements of the quotient group G/kerφ are the cosets of kerφ, i.e. all the squares.

Group theory is important because when counting “holes” in homology, G will be the group of cycles (the rubber bands).
The blue square will be the subgroup of “uninteresting rubber bands” that do not surround holes, similar to the earlier blue and green rubber bands. The quotient group “all rubber bands”/“uninteresting rubber bands” will identify holes.
However, the rubber bands are continuous and difficult to compute. We first need to discretize the space into a simpler structure called simplicial complex.





The intuition of simplicial complex is that if a simplex is in K, all its faces need to be in K, too. In addition, the simplices have to be glued together along
whole faces or be separate.
Simplicial complex plays the role of the yellow space in the
rubber band example. We next introduce the discrete version
of the rubber bands.

Definition 14. 
A p-chain is a subset of p-simplices in a simplicial complex K.

Definition 17. 
The boundary of a p-chain is the +2 sum of the boundaries of its simplices. Taking the boundary is a group homomorphism ∂p from Cp to Cp−1.
Note faces shared  by an even number of p-simplices in the chain will cancel  out:

the boundary of any higher order (p+1)-chain is always a p-cycle. For example, the left figure below shows a simplicial complex containing a (p+1) = 2 chain (the yellow triangle). Its boundary c1 (blue) is indeed a 1-cycle

Definition 19. 
A p-boundary-cycle is a p-cycle that is also the boundary of some (p+1)-chain.
Let Bp = ∂$_{p+1}$C$_{p+1}$, namely all the p-boundary-cycles.
Bp are the uninteresting rubber bands. In the example above, B1 = {0, c1}, none surrounding any holes.


c2,c3 are interesting because they surround the hole in the rectangle. In fact, we can drag the rubber band c2 over the yellow triangle and turn it into
c3. Formally, we do this by c3 = c2 + c1 . 
Intuitively, c2 and c3 are equivalent in the hole they surround. 
More generally, such equivalence class is obtained by c + Bp: we are allowed to drag a p-cycle rubber band c over any (p + 1)-simplices without changing the holes (or the lack thereof) it surrounds.
Returning to the example, we now see all the 1-cycles for this simplicial complex: Z1 = {0, c1, c2, c3}. The uninteresting ones are B1 = {0, c1}, a subgroup of Z1 . The interesting ones are c2+B1= c3+B1 = {c2, c3}: this should remind us of cosets and quotient group.(B1是子群H，操作符是+，c2+B1是coset，c3+B1也是coset；{c2, c3}是商群)

Definition 20. 
The p-th homology group is the quotient group Hp=Zp/Bp. The p-th Betti number is its rank: βp=rank(Hp).









*☆

 

 

 

 

 

 

 

 

 

Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing

Definition 4.
Given a subgroup H of an abelian group G, for any a∈G, the set a∗H = {a∗h | h∈H} is the coset of H represented by a.




Definition 7. 
The kernel of a homomorphism φ:G→G' is kerφ = {a∈G | φ(a)=e'}. 
In other words, the kernel is the elements that map to identity.


Define a new binary operation ☆ not on the elements of G but on the cosets of H:
(a∗H) ☆ (b∗H) = (a∗b)∗H, ∀a,b∈G.


Definition 8. 
The cosets {a∗H |a∈G} under the operation ☆ form a group, called the quotient group G/H.

quotient group G/H： <{a∗H |a∈G},☆>

- coset是一个set，(set里的元素是G里的元素)
- quotient group G/H里的每一个元素是(H的)一个coset



Persistent homology finds “holes” by identifying equivalent cycles: Consider the following space in yellow with a small white hole. Imagine the blue cycle as a rubber band. It can be stretched and bent within the space into the green cycle, but not the red one without tearing itself. 
There are two equivalent classes of rubber bands: some surround the hole and others do not. Conversely, two equivalent classes indicate one hole. 

Because kerφ is a subgroup, we can partition G into cosets of the form a∗kerφ for a∈G. These cosets are the white or blue squares.
It is useful to think of quotient groups as “higher level” groups defined on the squares in the previous picture. kerφ (the blue square) is a subgroup of G. The elements of the quotient group G/kerφ are the cosets of kerφ, i.e. all the squares.

Group theory is important because when counting “holes” in homology, G will be the group of cycles (the rubber bands).
The blue square will be the subgroup of “uninteresting rubber bands” that do not surround holes, similar to the earlier blue and green rubber bands. The quotient group “all rubber bands”/“uninteresting rubber bands” will identify holes.
However, the rubber bands are continuous and difficult to compute. We first need to discretize the space into a simpler structure called simplicial complex.





The intuition of simplicial complex is that if a simplex is in K, all its faces need to be in K, too. In addition, the simplices have to be glued together along
whole faces or be separate.
Simplicial complex plays the role of the yellow space in the
rubber band example. We next introduce the discrete version
of the rubber bands.

Definition 14. 
A p-chain is a subset of p-simplices in a simplicial complex K.

Definition 17. 
The boundary of a p-chain is the +2 sum of the boundaries of its simplices. Taking the boundary is a group homomorphism ∂p from Cp to Cp−1.
Note faces shared  by an even number of p-simplices in the chain will cancel  out:

the boundary of any higher order (p+1)-chain is always a p-cycle. For example, the left figure below shows a simplicial complex containing a (p+1) = 2 chain (the yellow triangle). Its boundary c1 (blue) is indeed a 1-cycle

Definition 19. 
A p-boundary-cycle is a p-cycle that is also the boundary of some (p+1)-chain.
Let Bp = ∂$_{p+1}$C$_{p+1}$, namely all the p-boundary-cycles.
Bp are the uninteresting rubber bands. In the example above, B1 = {0, c1}, none surrounding any holes.


c2,c3 are interesting because they surround the hole in the rectangle. In fact, we can drag the rubber band c2 over the yellow triangle and turn it into
c3. Formally, we do this by c3 = c2 + c1 . 
Intuitively, c2 and c3 are equivalent in the hole they surround. 
More generally, such equivalence class is obtained by c + Bp: we are allowed to drag a p-cycle rubber band c over any (p + 1)-simplices without changing the holes (or the lack thereof) it surrounds.
Returning to the example, we now see all the 1-cycles for this simplicial complex: Z1 = {0, c1, c2, c3}. The uninteresting ones are B1 = {0, c1}, a subgroup of Z1 . The interesting ones are c2+B1= c3+B1 = {c2, c3}: this should remind us of cosets and quotient group.(B1是子群H，操作符是+，c2+B1是coset，c3+B1也是coset；{c2, c3}是商群)

Definition 20. 
The p-th homology group is the quotient group Hp=Zp/Bp. The p-th Betti number is its rank: βp=rank(Hp).









*☆