About

Quickstart

The main function which we use is permaviss.spectral_sequence.MV_spectral_seq.create_MV_ss(). We start by taking 100 points in a noisy circle of radius 1

>>> from permaviss.sample_point_clouds.examples import random_circle
>>> point_cloud = random_circle(100, 1, epsilon=0.2)

Now we set the parameters for spectral sequence. These are

  • a prime number p,
  • the maximum dimension of the Rips Complex max_dim,
  • the maximum radius of filtration max_r,
  • the number of divisions max_div along the maximum range in point_cloud,
  • and the overlap between different covering regions.

In our case, we set the parameters to cover our circle with 9 covering regions. Notice that in order for the algorithm to give the correct result we need overlap > max_r.

>>> p = 3
>>> max_dim = 3
>>> max_r = 0.2
>>> max_div = 3
>>> overlap = max_r * 1.01

Then, we compute the spectral sequence, notice that the method prints the successive page ranks.

>>> from permaviss.spectral_sequence.MV_spectral_seq import create_MV_ss
>>> MV_ss = create_MV_ss(point_cloud, max_r, max_dim, max_div, overlap, p)
PAGE: 1
[[  0   0   0   0   0]
 [  7   0   0   0   0]
 [133  33   0   0   0]]
PAGE: 2
[[  0   0   0   0   0]
 [  7   0   0   0   0]
 [100   0   0   0   0]]
PAGE: 3
[[  0   0   0   0   0]
 [  7   0   0   0   0]
 [100   0   0   0   0]]
PAGE: 4
[[  0   0   0   0   0]
 [  7   0   0   0   0]
 [100   0   0   0   0]]

We can inspect the obtained barcodes on the 1st dimension.

>>> MV_ss.persistent_homology[1].barcode
array([[ 0.08218822,  0.09287436],
       [ 0.0874977 ,  0.11781674],
       [ 0.10459203,  0.12520266],
       [ 0.14999507,  0.18220508],
       [ 0.15036084,  0.15760192],
       [ 0.16260913,  0.1695936 ],
       [ 0.16462541,  0.16942819]])

Notice that in this case, there was no need to solve the extension problem. See the examples section for nontrivial extensions.

DISCLAIMER

The main purpose of this library is to explore how the Persistent Mayer Vietoris spectral sequence can be used for computing persistent homology.

This does not pretend to be an optimal library. Also, it does not parallelize the computations of persistent homology after the first page. Thus, this is slower than most other persistent homology computations.

This library is still on development and is still highly undertested. If you notice any issues, please email TorrasCasasA@cardiff.ac.uk

This library is published under the standard MIT licence. Thus: THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

How to cite

Álvaro Torras Casas. (2020, January 20). PerMaViss: Persistence Mayer Vietoris spectral sequence (Version v0.0.2). Zenodo. http://doi.org/10.5281/zenodo.3613870

Reference

This module is written using the algorithm in Distributing Persistent Homology via Spectral Sequences.