Choosing the parameter standard_range

The purpose of this example is to demonstrate the effect of the standard_range parameter.

For efficiency reasons, the algorithms implemented in this library work with Rips complexes; however, when this parameter is set to True, the output of the algorithms are guaranteed to represent non-trivial, continuous maps on the Čech complex of the data. In order to be able to guarantee this, the persistent cohomology class has to be sufficiently long, in the following sense: if the persistent cohomology class is represented by a point \((a,b)\) in the persistence diagram, we must have \(2a < b\). If \(2a \geq b\) and standard_range is set to True, the algorithm will return an error, and suggest setting standard_range to False.

The user can then decide to set standard_range to False, or to increase the parameter n_landmarks, which often results in a persistence diagram with classes that are born earlier (i.e., classes \((a,b)\) for which \(a\) is smaller).

import matplotlib.pyplot as plt
from dreimac import CircularCoords, GeometryExamples, CircleMapUtils
from persim import plot_diagrams

We consider two datasets, both consisting of a noisy circle with different levels of noise. We start with a not very noisy circle.

f, (a0, a1, a2) = plt.subplots(1, 3, width_ratios=[1, 1, 1], figsize=(14,3))

X1 = GeometryExamples.noisy_circle(n_samples = 200, noise_size=0.2)

a0.scatter(X1[:,0],X1[:,1], s = 10)
a0.set_title("Not very noisy circle") ; a0.axis("off") ; a0.set_aspect("equal")

cc = CircularCoords(X1, 30, prime=3)
plot_diagrams(cc._dgms, ax=a1)

circular_coordinates = cc.get_coordinates(standard_range=True)
a2.scatter(X1[:,0], X1[:,1], c=CircleMapUtils.to_sinebow(circular_coordinates), s=10)
a2.axis("off") ; a2.set_aspect("equal")

Now for a quite noisy circle.

f, (a0, a1) = plt.subplots(1, 2, width_ratios=[1, 1], figsize=(9,3))

X2 = GeometryExamples.noisy_circle(n_samples = 200, noise_size=0.8)

a0.scatter(X2[:,0],X2[:,1], s = 20)
a0.set_title("Very noisy circle") ; a0.axis("off") ; a0.set_aspect("equal")

cc2 = CircularCoords(X2, 30, prime=3)
plot_diagrams(cc2._dgms, ax=a1)

Using the most persistent class to construct a circular coordinate with standard_range equals to True results in an error, as described above:

try:
    circular_coordinates = cc2.get_coordinates(standard_range=True)
except Exception as e:
    print(e)

The cohomology class selected is too short, try setting standard_range to False.

However, using standard_range equals to False gives a good result, even if the theory does not guarantee this:

circular_coordinates2 = cc2.get_coordinates(standard_range=False)

plt.figure(figsize=(3,3))
plt.scatter(X2[:,0], X2[:,1], c=CircleMapUtils.to_sinebow(circular_coordinates2), s=20)
plt.gca().set_aspect("equal") ; _ = plt.axis("off")

As explained above, sometimes another option is to increase the parameter n_landmarks so that persistent cohomology classes are born earlier:

f, (a0, a1) = plt.subplots(1, 2, width_ratios=[1, 1], figsize=(9,3))

cc3 = CircularCoords(X2, 50, prime=3)
plot_diagrams(cc._dgms, ax=a0)

circular_coordinates3 = cc3.get_coordinates(standard_range=True)
a1.scatter(X2[:,0], X2[:,1], c=CircleMapUtils.to_sinebow(circular_coordinates3), s=20)
a1.axis("off") ; a1.set_aspect("equal")