Notebook source code: notebooks/03_practical_methods__data_on_manifolds.ipynb
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Data on Manifolds#

Lead author: Nina Miolane.

We import the backend that will be used for geomstats computations and set a seed for reproducibility of the results.

 In [1]:

import geomstats.backend as gs

gs.random.seed(2020)

INFO: Using numpy backend

Finally, we import the visualization module.

 In [2]:

import os

import matplotlib.colors as colors
import matplotlib.image as mpimg
import matplotlib.patches as mpatches
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D  # NOQA
from mpl_toolkits.mplot3d.art3d import Poly3DCollection

import geomstats.visualization as visualization

visualization.tutorial_matplotlib()

From data on linear spaces to data on manifolds#

The science of Statistics is defined as the collection of data, their analysis and interpretation. Statistical theory is usually defined for data belonging to vector spaces, which are linear spaces. For example, we know how to compute the mean of a data set of numbers, like the mean of students’ weights in a classroom, or of multidimensional arrays, like the average 3D velocity vector of blood cells in a vessel.

Here is an example of the computation of the mean of two arrays of dimension 2.

 In [3]:

from geomstats.geometry.euclidean import Euclidean

dim = 2
n_samples = 2

euclidean = Euclidean(dim=dim, equip=False)
points_in_linear_space = euclidean.random_point(n_samples=n_samples)
print("Points in linear space:\n", points_in_linear_space)

linear_mean = gs.sum(points_in_linear_space, axis=0) / n_samples
print("Mean of points:\n", linear_mean)

Points in linear space:
 [[ 0.97255366  0.74678389]
 [ 0.01949105 -0.45632857]]
Mean of points:
 [0.49602235 0.14522766]

We plot the points and their mean on the 2D Euclidean space, which is a linear space: a plane.

 In [4]:

%matplotlib inline

fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(111)

ax.scatter(points_in_linear_space[:, 0], points_in_linear_space[:, 1], label="Points")
ax.plot(points_in_linear_space[:, 0], points_in_linear_space[:, 1], linestyle="dashed")

ax.scatter(
    gs.to_numpy(linear_mean[0]),
    gs.to_numpy(linear_mean[1]),
    label="Mean",
    s=80,
    alpha=0.5,
)

ax.set_title("Mean of points in a linear space")
ax.legend();

../_images/notebooks_03_practical_methods__data_on_manifolds_10_0.png

What happens to the usual statistical theory when the data doesn’t naturally belong to a linear space. For example, if we want to perform statistics on the coordinates of world cities, which lie on the earth: a sphere?

The non-linear spaces we consider are called manifolds. A manifold \(M\) of dimension \(m\) is a space that is allowed to be curved but that looks like an \(m\)-dimensional vector space in the neighborhood of every point.

A sphere, like the earth, is a good example of a manifold. We know that the earth is curved, but at our scale we do not see its curvature. Can we still use linear statistics when data are defined on these manifolds, or shall we?

Let’s try.

 In [5]:

from geomstats.geometry.hypersphere import Hypersphere

sphere = Hypersphere(dim=dim, equip=False)
points_in_manifold = sphere.random_uniform(n_samples=n_samples)
print("Points in manifold:\n", points_in_manifold)

linear_mean = gs.sum(points_in_manifold, axis=0) / n_samples
print("Mean of points:\n", linear_mean)

Points in manifold:
 [[-0.71502435 -0.41197257 -0.56481748]
 [-0.997575   -0.04788171  0.05051201]]
Mean of points:
 [-0.85629967 -0.22992714 -0.25715273]

We plot the points and their mean computed with the linear formula.

 In [6]:

%matplotlib inline

fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111, projection="3d")

visualization.plot(points_in_manifold, ax=ax, space="S2", label="Point", s=80)

ax.plot(
    points_in_manifold[:, 0],
    points_in_manifold[:, 1],
    points_in_manifold[:, 2],
    linestyle="dashed",
    alpha=0.5,
)

ax.scatter(
    linear_mean[0], linear_mean[1], linear_mean[2], label="Mean", s=80, alpha=0.5
)

ax.set_title("Mean of points on a manifold")
ax.legend();

../_images/notebooks_03_practical_methods__data_on_manifolds_14_0.png

What happened? The mean of two points on a manifold (the sphere) is not on the manifold. In our example, the mean city is not on the earth. This leads to errors in statistical computations.

 In [7]:

print(sphere.belongs(linear_mean))

False

For this reason, researchers aim to build a theory of statistics that is by construction compatible with any structure we equip the manifold with. This theory is called Geometric Statistics.

Geometric Statistics is a theory of statistics on manifolds, that takes into account their geometric structures. Geometric Statistics is therefore the child of two major pillars of Mathematics: Geometry and Statistics.

Examples of data on manifolds#

Why should we bother to build a whole new theory of statistics? Do we really have data that belong to spaces like the sphere illustrated in introduction?

Let’s see some examples of data spaces that are naturally manifolds. By doing so, we will introduce the datasets and visualization modules of geomstats.

We first import the datasets.utils module that allows loading datasets.

 In [8]:

import geomstats.datasets.utils as data_utils

World cities: data on the sphere#

We load the dataset cities, that contains the coordinates of world cities in spherical coordinates.

 In [9]:

data, names = data_utils.load_cities()
print(names[:5])
print(data[:5])

['Tokyo', 'New York', 'Mexico City', 'Mumbai', 'São Paulo']
[[ 0.61993792 -0.52479018  0.58332859]
 [-0.20994315  0.7285533   0.65202298]
 [ 0.14964311  0.93102728  0.33285904]
 [-0.27867026 -0.9034188   0.32584868]
 [-0.62952884  0.6662902  -0.3996884 ]]

We convert the spherical coordinates to X, Y, Z coordinates and verify that they belong to the sphere.

 In [10]:

gs.all(sphere.belongs(data))

 Out [10]:

True

Now, we plot the cities on the sphere. We choose only a subset of the cities that have a nice rendering in the 2D plot of the 3D earth. This plot is nicer shown in an interactive 3D figure.

 In [11]:

data, names = data_utils.load_cities()

 In [12]:

fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, projection="3d")

visualization.plot(data[15:20], ax=ax, space="S2", label=names[15:20], s=80, alpha=0.5)

ax.set_title("Cities on the earth.");

../_images/notebooks_03_practical_methods__data_on_manifolds_28_0.png

Pose of objects in pictures: data on the Lie group of 3D rotations#

We consider the dataset poses, that contains the 3D poses of objects in images. Specifically, we consider poses of beds in images, i.e. the 3D orientation of each bed within a given 2D image.

The orientation corresponds to a 3D rotation. A 3D rotation \(R\) is visually represented as the result of \(R\) applied to the coordinate frame \((e_x, e_y, e_z)\).

We first load the dataset.

 In [13]:

data, img_paths = data_utils.load_poses()

img_path1, img_path2 = img_paths[0], img_paths[1]
img_path1 = os.path.join(data_utils.DATA_PATH, "poses", img_path1)
img_path2 = os.path.join(data_utils.DATA_PATH, "poses", img_path2)

img1 = mpimg.imread(img_path1)
img2 = mpimg.imread(img_path2)

fig = plt.figure(figsize=(16, 8))

ax = fig.add_subplot(121)
imgplot = ax.imshow(img1)
ax.axis("off")
ax = fig.add_subplot(122)
imgplot = ax.imshow(img2)
ax.axis("off");

../_images/notebooks_03_practical_methods__data_on_manifolds_31_0.png

We import the manifold of 3D rotations: the Special Orthogonal group in 3D, \(SO(3)\). We choose to represent the 3D rotations as rotation vectors, hence: point_type='vector'.

 In [14]:

from geomstats.geometry.special_orthogonal import SpecialOrthogonal

so3 = SpecialOrthogonal(n=3, point_type="vector", equip=False)

We verify that the poses belong to the space of 3D rotations.

 In [15]:

gs.all(so3.belongs(data))

 Out [15]:

True

We plot the orientations of the first 2 beds.

 In [16]:

fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111, projection="3d")

visualization.plot(data[:2], ax=ax, space="SO3_GROUP")

ax.set_title("3D orientations of the beds.");

../_images/notebooks_03_practical_methods__data_on_manifolds_37_0.png

These orientations are very close, as expected from the corresponding images.

Brain connectomes: data on the manifold of Symmetric Positive Definite (SPD) matrices#

We consider the dataset connectomes that contains brain functional connectomes from the MSLP 2014 Schizophrenia Challenge. The dataset correponds to the Functional Connectivity Networks (FCN) extracted from resting-state fMRIs of 86 patients at 28 Regions Of Interest (ROIs). Roughly, an FCN corresponds to a correlation matrix and can be seen as a point on the manifold of Symmetric Positive-Definite (SPD) matrices.

We can load the dataset.

 In [20]:

data, patient_ids, labels = data_utils.load_connectomes()

We plot the first two connectomes from the MSLP dataset with their corresponding labels.

 In [21]:

labels_str = ["Healthy", "Schizophrenic"]

fig = plt.figure(figsize=(8, 4))

ax = fig.add_subplot(121)
imgplot = ax.imshow(data[0])
ax.set_title(labels_str[labels[0]])

ax = fig.add_subplot(122)
imgplot = ax.imshow(data[1])
ax.set_title(labels_str[labels[1]]);

../_images/notebooks_03_practical_methods__data_on_manifolds_51_0.png

We check that the connectomes belong to the space of SPD matrices.

 In [22]:

import geomstats.geometry.spd_matrices as spd

manifold = spd.SPDMatrices(28, equip=False)

 In [23]:

gs.all(manifold.belongs(data))

 Out [23]:

True

Monkey’s optical nerve heads: Data as landmarks in 3D#

We consider 22 images of Rhesus monkeys’ eyes (11 monkeys), acquired with a Heidelberg Retina Tomograph and available in (PE2015). For each monkey, an experimental glaucoma was introduced in one eye, while the second eye was kept as control. One seeks a significant difference between the glaucoma and the control eyes. On each image, 5 anatomical landmarks were recorded: - 1st landmark: superior aspect of the retina, - 2nd landmark: side of the retina closest to the temporal bone of the skull, - 3rd landmark: nose side of the retina, - 4th landmark: inferior point, - 5th landmark: optical nerve head deepest point.

Label 0 refers to a normal eye, and Label 1 to an eye with glaucoma.

Reference:#

(PE2015) Patrangenaru and L. Ellingson. Nonparametric Statistics on Manifolds and Their Applications to Object Data, 2015. https://doi.org/10.1201/b18969

We load the dataset of landmarks’ sets and corresponding labels.

 In [24]:

nerves, labels, monkeys = data_utils.load_optical_nerves()
print(nerves.shape)
print(labels)
print(monkeys)

(22, 5, 3)
[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
[ 0  0  1  1  2  2  3  3  4  4  5  5  6  6  7  7  8  8  9  9 10 10]

We extract the landmarks’ sets corresponding to the two eyes’ nerves of the first monkey, with their corresponding labels.

 In [25]:

two_nerves = nerves[monkeys == 0]
print(two_nerves.shape)

two_labels = labels[monkeys == 0]
print(two_labels)

(2, 5, 3)
[0 1]

 In [26]:

label_to_str = {0: "Normal nerve", 1: "Glaucoma nerve"}
label_to_color = {
    0: (102 / 255, 178 / 255, 255 / 255, 1.0),
    1: (255 / 255, 178 / 255, 102 / 255, 1.0),
}

ax = plt.figure(figsize=(8, 8)).add_subplot(projection="3d")

ax.set_xlim((2000, 4000))
ax.set_ylim((1000, 5000))
ax.set_zlim((-600, 200))

for nerve, label in zip(two_nerves, two_labels):
    x = nerve[:, 0]
    y = nerve[:, 1]
    z = nerve[:, 2]

    verts = [list(zip(x, y, z))]

    poly = Poly3DCollection(verts, alpha=0.5)
    color = label_to_color[int(label)]
    poly.set_color(colors.rgb2hex(color))
    poly.set_edgecolor("k")
    ax.add_collection3d(poly)

patch_0 = mpatches.Patch(color=label_to_color[0], label=label_to_str[0], alpha=0.5)
patch_1 = mpatches.Patch(color=label_to_color[1], label=label_to_str[1], alpha=0.5)
plt.legend(handles=[patch_0, patch_1], prop={"size": 20});

../_images/notebooks_03_practical_methods__data_on_manifolds_61_0.png

We observe that the shape of the optical nerve head on the eye with glaucoma is different from the shape of the optical nerve head on the normal eye.

Data on Manifolds#

From data on linear spaces to data on manifolds#

Examples of data on manifolds#

World cities: data on the sphere#

Pose of objects in pictures: data on the Lie group of 3D rotations#

Social networks: data on the hyperbolic space#

Brain connectomes: data on the manifold of Symmetric Positive Definite (SPD) matrices#

Monkey’s optical nerve heads: Data as landmarks in 3D#

Reference:#

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