How to interpret homology_persistence output

[grjd]

The output of homology_persistence is not quite self explanatory. It would be nice to have a mre informative message for example, stating the birth and death of the homology class.
In my example, I use a filtration starting with 4 disconnected nodes, ending up forming a tetrahedron. What the output means?

simplices = [([0],1), ([1],2),([2],3),([3],4),([0,1],5),([2,3],6), ([1,2],7),([1,3],8), ([0,2],9), ([0,3],10)]
f =d.Filtration()
m = d.homology_persistence(f)
for c,i in enumerate(m):
   print(c,i)

(0, )
(1, )
(2, )
(3, )
(4, 10 + 11)
(5, 12 + 13)
(6, 11 + 12)
(7, )
(8, )
(9, )

[mrzv]

You are iterating over the internal representation of the reduced boundary matrix (which is giving you cycles that are killed by the simplices, or empty cycles if a class if born). You probably want to get a persistence diagram. More details are at http://mrzv.org/software/dionysus2/tutorial/basics.html

[sauln]

Hi @mrzv, I'm also having trouble interpreting this output. In the basics page you link, an example is

 >>> m = d.homology_persistence(f)
 >>> for i,c in enumerate(m):
 ...     print(i, c)
 0
 1
 2 1*0 + 1*1
 3
 4 1*1 + 1*3
 5

Could you please explain what each of the numbers in the output mean:

i a*b + c*d

I am looking to compute representative cycles and am hoping this information will give it to me.

Thank you.

[mrzv]

m is the reduced boundary matrix. If we refer to the simplices in the filtration order as s_0, s_1, s_2, ... then

i a*b + c*d

means that addition of simplex s_i sent (previously non-zero) cycle a*s_b + c*s_d to zero.

Does that make sense?

[mrzv]

I should add that depending on which optimization you use, the cycles might be pruned (i.e., missing some simplices). This shouldn't happen by default, but it will happen, if you pass, for example, method = "column_no_negative".