[1]:

# To run this in Google Colab, uncomment the following line
# !pip install geometric_kernels

# If you want to use a version of the library from a specific branch on GitHub,
# say, from the "devel" branch, uncomment the line below instead
# !pip install "git+https://github.com/geometric-kernels/GeometricKernels@devel"

Matérn and Heat Kernels on Spheres¶

This notebook shows how define and evaluate kernels on the two-dimensional sphere $\mathbb{S}_2 \subseteq \mathbb{R^3}$.

Handling hyperspheres $\mathbb{S}_d \subseteq \mathbb{R}^{d+1}$ for $d \geq 2$ is the same. We chose to showcase $\mathbb{S}_2$ here because (1) it is probably the most well known case (2) it is easy to visualize.

Note: the points on the (hyper)sphere $\mathbb{S}_d$ are represented by vectors (arrays of the suitable backend) in $\mathbb{R}^{d+1}$ whose Euclidean norm is equal to $1$.

We use the numpy backend here.

Contents¶

Basics
Feature Maps and Sampling
- Defining a Feature Map
- Efficient Sampling using Feature Maps
Citation

Basics¶

[2]:

# Import a backend, we use numpy in this example.
import numpy as np

# Import the geometric_kernels backend.
import geometric_kernels

# Note: if you are using a backend other than numpy,
# you _must_ uncomment one of the following lines
# import geometric_kernels.tensorflow
# import geometric_kernels.torch
# import geometric_kernels.jax

# Import a space and an appropriate kernel.
from geometric_kernels.spaces import Hypersphere
from geometric_kernels.kernels import MaternGeometricKernel

import matplotlib as mpl
import matplotlib.pyplot as plt

INFO (geometric_kernels): Numpy backend is enabled, as always. To enable other backends, don't forget `import geometric_kernels.*backend name*`.

Defining a Space¶

First we create a GeometricKernels space that corresponds to the 2-dimensional sphere $\mathbb{S}_2$, a subset of $\mathbb{R}^3$.

[3]:

sphere = Hypersphere(dim=2)

Defining a Kernel¶

First, we create a generic Matérn kernel.

To initialize MaternGeometricKernel you just need to provide a Space object, in our case this is the sphere we have just created above.

There is also an optional second parameter num which determines the order of approximation of the kernel. There is a sensible default value for each of the spaces in the library, so change it only if you know what you are doing.

A brief account on theory behind the kernels on compact manifolds like (which hyperspheres are examples of) can be found on these documentation pages: one, two.

[4]:

kernel = MaternGeometricKernel(sphere)

To support JAX, our classes do not keep variables you might want to differentiate over in their state. Instead, some methods take a params dictionary as input, returning its modified version.

The next line initializes the dictionary of kernel parameters params with some default values.

Note: our kernels do not provide the outputscale/variance parameter frequently used in Gaussian processes. However, it is usually trivial to add it by multiplying the kernel by an (optimizable) constant.

[5]:

params = kernel.init_params()
print('params:', params)

params: {'lengthscale': array(1.), 'nu': array(inf)}

To define two different kernels, Matern-3/2 and Matern-∞ (aka heat, RBF, squared exponential, diffusion), we need two different versions of params:

[6]:

params["lengthscale"] = np.array([0.125])
params_32  = params.copy()
params_inf = params.copy()
del params
params_32["nu"]  = np.array([3/2])
params_inf["nu"] = np.array([np.inf])

Now two kernels are defined and we proceed to evaluating both on a set of random inputs.

Evaluating Kernels on Random Inputs¶

We start by sampling 10 (uniformly) random points on the sphere $\mathbb{S}_2$. An explicit key parameter is needed to support JAX as one of the backends.

[7]:

key = np.random.RandomState(1234)

key, xs = sphere.random(key, 10)

print(xs)

[[ 0.2453089  -0.61971816  0.74550181]
 [-0.26385701 -0.60812805  0.74870539]
 [ 0.80356397 -0.59503751  0.01467335]
 [-0.8280059   0.42459658  0.3662294 ]
 [ 0.42189482 -0.89450892 -0.14784637]
 [ 0.00425403  0.81421747  0.58054441]
 [ 0.64624279 -0.75666681 -0.09912418]
 [-0.74561122  0.21985349  0.62906943]
 [ 0.84836694 -0.30204652  0.43478896]
 [-0.86072199 -0.08673813  0.50163149]]

Now we evaluate the two kernel matrices.

[8]:

kernel_mat_32  = kernel.K(params_32,  xs, xs)
kernel_mat_inf = kernel.K(params_inf, xs, xs)

Finally, we visualize these matrices using imshow.

[9]:

# find common range of values
minmin = np.min([np.min(kernel_mat_32), np.min(kernel_mat_inf)])
maxmax = np.max([np.max(kernel_mat_32), np.max(kernel_mat_inf)])

fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2)
cmap = plt.get_cmap('viridis')

ax1.imshow(kernel_mat_32, vmin=minmin, vmax=maxmax, cmap=cmap)
ax1.set_title('k_32')
ax1.set_axis_off()

ax2.imshow(kernel_mat_inf, vmin=minmin, vmax=maxmax, cmap=cmap)
ax2.set_title('k_inf')
ax2.set_axis_off()

# add space for color bar
fig.subplots_adjust(right=0.85)
cbar_ax = fig.add_axes([0.88, 0.25, 0.02, 0.5])

# add colorbar
sm = plt.cm.ScalarMappable(cmap=cmap,
                           norm=plt.Normalize(vmin=minmin, vmax=maxmax))
fig.colorbar(sm, cax=cbar_ax)

plt.show()

../_images/examples_Hypersphere_22_0.png

Visualizing Kernels¶

The sphere $\mathbb{S}_2$ is one of the few manifolds we can easily visualize. Because of this, beyond kernel matrices, we will plot the functions $k_{\nu, \kappa}(\text{pole}, \cdot)$.

In practice, we visualize $k_{\nu, \kappa}($ base_point $, x)$ for $x \in `$ ``other_points`. The base_point is simply the north pole. The other_points is defined by the _NUM_LATS x _NUM_LONGS uniform grid.

We define base_point and other_points in the next cell.

[10]:

base_point = np.array([[0.0, 0.0, 1.0]])

# define latitude and longitude discretization
_NUM_LATS = 128
_NUM_LONGS = 128

# generate a grid on the sphere
lats, longs =  np.mgrid[0:2*np.pi:_NUM_LATS*1j, 0:np.pi:_NUM_LONGS*1j]

other_points_xs = np.sin(longs) * np.cos(lats)
other_points_ys = np.sin(longs) * np.sin(lats)
other_points_zs = np.cos(longs)

other_points = np.c_[np.ravel(other_points_xs),
                     np.ravel(other_points_ys),
                     np.ravel(other_points_zs)]

The next cell evaluates $k_{\nu, \kappa}($ base_point $, x)$ for $x \in `$ ``other_points` for $\nu$ either $3/2$ or $\infty$.

[11]:

kernel_vals_32  = kernel.K(params_32,  base_point, other_points)
kernel_vals_inf = kernel.K(params_inf, base_point, other_points)

Finally, we are ready to plot the results.

[12]:

fig, (ax1, ax2) = plt.subplots(figsize=(12.8, 4.8), nrows=1, ncols=2,
                               subplot_kw=dict(projection='3d',
                                               computed_zorder=False))

cmap = plt.get_cmap('viridis')

surf1 = ax1.plot_surface(other_points_xs, other_points_ys, other_points_zs,
                         facecolors=cmap(kernel_vals_32[0, :].reshape(_NUM_LATS, _NUM_LONGS)),  cstride=1, rstride=1)
surf2 = ax2.plot_surface(other_points_xs, other_points_ys, other_points_zs,
                         facecolors=cmap(kernel_vals_inf[0, :].reshape(_NUM_LATS, _NUM_LONGS)), cstride=1, rstride=1)

# Remove axis
ax1._axis3don = False
ax2._axis3don = False

# Set aspect ratio
ax1.set_box_aspect((np.ptp(other_points_xs), np.ptp(other_points_ys),
                    np.ptp(other_points_zs)), zoom=1.5)
ax2.set_box_aspect((np.ptp(other_points_xs), np.ptp(other_points_ys),
                    np.ptp(other_points_zs)), zoom=1.5)

# plot the base point
ax1.scatter(base_point[0, 0], base_point[0, 1], base_point[0, 2], s=10, c='r', marker='D')
ax2.scatter(base_point[0, 0], base_point[0, 1], base_point[0, 2], s=10, c='r', marker='D')

# find common range of values
minmin_vis = np.min([np.min(kernel_vals_32), np.min(kernel_vals_inf)])
maxmax_vis = np.max([np.max(kernel_vals_32), np.max(kernel_vals_inf)])

ax1.set_title(r'$k_{3/2, \kappa}($base_point, other_points$)$')
ax2.set_title('$k_{\infty, \kappa}($base_point, other_points$)$')

# add color bar
sm = plt.cm.ScalarMappable(cmap=cmap,
                           norm=plt.Normalize(vmin=minmin_vis, vmax=maxmax_vis))
fig.subplots_adjust(right=0.85)
cbar_ax = fig.add_axes([0.88, 0.25, 0.02, 0.5])
fig.colorbar(sm, cax=cbar_ax)

plt.show()

../_images/examples_Hypersphere_29_0.png

Feature Maps and Sampling¶

Here we show how to get an approximate finite-dimensional feature map for heat and Matérn kernels on the sphere, i.e. such $\phi$ that

\[k(x, x') \approx \langle \phi(x), \phi(x') \rangle_{\mathbb{R}^M}.\]

This might be useful for speeding up computations. We showcase this below by showing how to efficiently sample the Gaussian process $\mathrm{GP}(0, k)$.

For a brief theoretical introduction into feature maps, see this documentation page.

Defining a Feature Map¶

The simplest way to get an approximate finite-dimensional feature map is to use the default_feature_map function from geometric_kernels.kernels. It has an optional keyword argument num which determines the number of features, the $M$ above. Below we rely on the default value of num.

[13]:

from geometric_kernels.kernels import default_feature_map

feature_map = default_feature_map(kernel=kernel)

The resulting feature_map is a function that takes the array of inputs and parameters of the kernel. There is also an optional parameter normalize that determines if $\langle \phi(x), \phi(x) \rangle_{\mathbb{R}^M} \approx 1$ or not. For the (hyper)sphere, normalize follows the standard behavior of MaternKarhunenLoeveKernel, being True by default.

feature_map outputs a tuple. Its second element is $\phi(x)$ evaluated at all inputs $x$. Its first element is either None for determinstic feature maps, or contains the updated key for randomized feature maps which take key as a keyword argument. For default_feature_map on a Hypersphere space, the first element is None since the feature map is deterministic.

In the next cell, we evaluate the feature map at random points, using params_32 as kernel parameters. We check the basic property of the feature map: $k(x, x') \approx \langle \phi(x), \phi(x') \rangle_{\mathbb{R}^M}$.

[14]:

# xs are random points from above
_, embedding = feature_map(xs, params_32)

print('xs (shape = %s):\n%s' % (xs.shape, xs))
print('')
print('emedding (shape = %s):\n%s' % (embedding.shape, embedding))

kernel_mat_32  = kernel.K(params_32,  xs, xs)
kernel_mat_32_alt = np.matmul(embedding, embedding.T)

print('')
print('||k(xs, xs) - phi(xs) * phi(xs)^T|| =', np.linalg.norm(kernel_mat_32 - kernel_mat_32_alt))

xs (shape = (10, 3)):
[[ 0.2453089  -0.61971816  0.74550181]
 [-0.26385701 -0.60812805  0.74870539]
 [ 0.80356397 -0.59503751  0.01467335]
 [-0.8280059   0.42459658  0.3662294 ]
 [ 0.42189482 -0.89450892 -0.14784637]
 [ 0.00425403  0.81421747  0.58054441]
 [ 0.64624279 -0.75666681 -0.09912418]
 [-0.74561122  0.21985349  0.62906943]
 [ 0.84836694 -0.30204652  0.43478896]
 [-0.86072199 -0.08673813  0.50163149]]

emedding (shape = (10, 625)):
[[ 0.09376086  0.11951028  0.08744111 ... -0.00157167  0.00251287
  -0.00401033]
 [ 0.09376086  0.12002385  0.01822028 ... -0.0063809  -0.00453905
  -0.00676846]
 [ 0.09376086  0.00235226  0.16004253 ...  0.00755857 -0.0211777
  -0.01170085]
 ...
 [ 0.09376086  0.10084518 -0.11923475 ...  0.00760885  0.00556771
   0.00056388]
 [ 0.09376086  0.06970037  0.14023769 ...  0.0068301  -0.00032934
   0.0070823 ]
 [ 0.09376086  0.0804158  -0.10764934 ...  0.00091627 -0.01144613
  -0.00911491]]

||k(xs, xs) - phi(xs) * phi(xs)^T|| = 3.139317061076477e-07

Efficient Sampling using Feature Maps¶

GeometricKernels provides a simple tool to efficiently sample (without incurring cubic costs) the Gaussian process $f \sim \mathrm{GP}(0, k)$, based on an approximate finite-dimensional feature map $\phi$. The underlying machinery is briefly discussed in this documentation page.

The function sampler from geometric_kernels.sampling takes in a feature map and, optionally, the keyword argument s that specifies the number of samples to generate. It returns a function we name sample_paths.

sample_paths operates much like feature_map above: it takes in the points where to evaluate the samples and kernel parameters. Additionally, it takes in the keyword argument key that specifies randomness in the JAX style. sample_paths returns a tuple. Its first element is the updated key. Its second element is an array containing the value of samples evaluated at the input points.

[15]:

from geometric_kernels.sampling import sampler

sample_paths = sampler(feature_map, s=2)

# introduce random state for reproducibility (optional)
# `key` is jax's terminology
key = np.random.RandomState(seed=1234)

# new random state is returned along with the samples
key, samples = sample_paths(xs, params_32, key=key)

print('Two samples evaluated at the xs are:')
print(samples)

Two samples evaluated at the xs are:
[[-1.29069161 -0.01041011]
 [ 2.01336997 -1.95191625]
 [-1.49327597  1.92384502]
 [ 1.31761287 -0.22536152]
 [-1.49768137  1.18209194]
 [ 0.78375713  0.66272087]
 [ 0.80986794  0.71381113]
 [-0.61199444  1.25119568]
 [-0.91047851  0.08939734]
 [-0.62337924 -1.07361345]]

Visualizing Samples¶

Here we visualize samples as functions on the sphere.

[16]:

key = np.random.RandomState(seed=1234)
key, samples = sample_paths(other_points, params_32, key=key)

sample1 = samples[:, 0]
sample2 = samples[:, 1]

fig, (ax1, ax2) = plt.subplots(figsize=(12.8, 4.8), nrows=1, ncols=2,
                               subplot_kw=dict(projection='3d',
                                               computed_zorder=False))

cmap = plt.get_cmap('viridis')

surf1 = ax1.plot_surface(other_points_xs, other_points_ys, other_points_zs,
                         facecolors=cmap(sample1.reshape(_NUM_LATS, _NUM_LONGS)),  cstride=1, rstride=1)
surf2 = ax2.plot_surface(other_points_xs, other_points_ys, other_points_zs,
                         facecolors=cmap(sample2.reshape(_NUM_LATS, _NUM_LONGS)), cstride=1, rstride=1)

# Remove axis
ax1._axis3don = False
ax2._axis3don = False

# Set aspect ratio
ax1.set_box_aspect((np.ptp(other_points_xs), np.ptp(other_points_ys),
                    np.ptp(other_points_zs)), zoom=1.5)
ax2.set_box_aspect((np.ptp(other_points_xs), np.ptp(other_points_ys),
                    np.ptp(other_points_zs)), zoom=1.5)

# find common range of values
minmin_vis = np.min([np.min(sample1), np.min(sample2)])
maxmax_vis = np.max([np.max(sample1), np.max(sample2)])

ax1.set_title('Sample #1: $f(\cdot)$ for $\cdot$ in other_points, $f \sim \mathrm{GP}(0, k_{3/2, \kappa})$')
ax2.set_title('Sample #2: $f(\cdot)$ for $\cdot$ in other_points, $f \sim \mathrm{GP}(0, k_{3/2, \kappa})$')

# add color bar
sm = plt.cm.ScalarMappable(cmap=cmap,
                           norm=plt.Normalize(vmin=minmin_vis, vmax=maxmax_vis))
fig.subplots_adjust(right=0.85)
cbar_ax = fig.add_axes([0.88, 0.25, 0.02, 0.5])
fig.colorbar(sm, cax=cbar_ax)
plt.show()

../_images/examples_Hypersphere_39_0.png

Citation¶

If you are using (hyper)spheres and GeometricKernels, please consider citing

@article{mostowsky2024,
      title = {The GeometricKernels Package: Heat and Matérn Kernels for Geometric Learning on Manifolds, Meshes, and Graphs},
      author = {Peter Mostowsky and Vincent Dutordoir and Iskander Azangulov and Noémie Jaquier and Michael John Hutchinson and Aditya Ravuri and Leonel Rozo and Alexander Terenin and Viacheslav Borovitskiy},
      year = {2024},
      journal = {arXiv:2407.08086},
}

@inproceedings{borovitskiy2020,
    title={Matérn Gaussian processes on Riemannian manifolds},
    author={Viacheslav Borovitskiy and Alexander Terenin and Peter Mostowsky and Marc Peter Deisenroth},
    booktitle={Advances in Neural Information Processing Systems},
    year={2020}
}

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