permaviss.persistence_algebra.image_kernel

image_kernel.py

This module implements a function which computes bases for the image and kernel of morphisms between persistence modules.

Functions

image_kernel(A, B, F, p[, start_index, …]) This computes basis for the image and kernel of a persistence morphism.
permaviss.persistence_algebra.image_kernel._pivot(l)[source]
Parameters:l (list(int)) – List of integers to compute pivot from.
Returns:Index of last nonzero entry on l. Returns -1 if the list is zero.
Return type:int
permaviss.persistence_algebra.image_kernel.image_kernel(A, B, F, p, start_index=0, prev_basis=None)[source]
This computes basis for the image and kernel of a persistence morphism.
f: A –> B

This is the algorithm described in https://arxiv.org/abs/1907.05228. Recall that for such an algorithm to work the A and B must be ordered. This is why the function first orders the barcode generators from start_index until A.dim. Additionally, it also orders the barcodes from B. By ‘ordered’ we mean that the barcodes are sorted according to the standard order of barcodes.

It can also compute relative barcode bases for the image. This is used when computing quotients. The optional argument start_index indicates the minimum index from which we want to compute barcodes relative to the previous generators. That is, given start_index, the function will return image barcodes for

<F[start_dim, …, A.dim]> mod <F[0,1,…, start_dim-1]>.

At the end the bases for the image and kernel are returned in terms of the original ordering.

Additionally, this handles the case for when B is a broken barcode basis. Notice that in such a case, only the barcode basis of the image will be computed

Parameters:
  • A (barcode_basis object) – Basis of domain.
  • B (barcode_basis object) – Basis of range. This can be a broken barcode basis.
  • F (Numpy Array (B.dim, A.dim)) – Matrix associated to the considered persistence morphism.
  • p (int) – prime number of finite field.
  • start_index (int, default is 0) – Index from which we get a barcode basis for the image.
  • prev_basis (int, default is None) – If start_index > 0, we need to also give a reference to a basis of barcodes from A[start_dim] until A[A.dim].
Returns:

  • Ker (barcode_basis object) – Absolute/relative basis of kernel of f.
  • Im (barcode_basis object) – Absolute/relative basis of image of f.
  • PreIm (Numpy Array (A.dim, Im.dim)) – Absolute/relative preimage coordinates of f. That is, each column stores the sums that generate the corresponding Image barcode.

Examples

>>> import numpy as np
>>> from permaviss.persistence_algebra.barcode_bases import
... barcode_basis
>>> A = barcode_basis([[1,8],[1,5],[2,5], [4,8]])
>>> B = barcode_basis([[-1,3],[0,4],[0,3.5],[2,5],[2,4],[3,8]])
>>> F = np.array([[4,1,1,0],[1,4,1,0],[1,1,4,0],[0,0,1,4],[0,0,4,1],
... [0,0,0,1]])
>>> p = 5
>>> Im, Ker, PreIm = image_kernel(A,B,F,p)
>>> print(Im)
Barcode basis
[[ 1.   4. ]
 [ 1.   3.5]
 [ 2.   5. ]
 [ 4.   8. ]]
[[ 4.  0.  1.  0.]
 [ 1.  0.  1.  0.]
 [ 1.  2.  4.  0.]
 [ 0.  0.  1.  4.]
 [ 0.  0.  4.  1.]
 [ 0.  0.  0.  1.]]
>>> print(Ker)
Barcode basis
[[ 3.5  8. ]
 [ 4.   5. ]]
[[ 1.  0.]
 [ 1.  4.]
 [ 0.  0.]
 [ 0.  0.]]
>>> print(PreIm)
[[ 1.  1.  0.  0.]
 [ 0.  1.  0.  0.]
 [ 0.  0.  1.  0.]
 [ 0.  0.  0.  1.]]

Note

This algorithm will only work if the matrix of the persistence morphism is well defined. That is, a generator can only map to generators that have been born and have not yet died.