SphereDiff

Purpose

The purpose of this package is to sample from the isotropic diffusion kernel on the surface of an n-dimensional unit sphere, where n is an arbitrary natural number dimension greater than or equal to 3.

Installation

From a repository checkout

pip install --user .

From PyPI

pip install --user spherediff

Use

The user may sample from the isotropic Gaussian distribution on the unit n-sphere using the sample_spherical_kernel function, which may be imported as follows:

>> from spherediff.sample import sample_spherical_kernel

This function takes three arguments and one additional, optional argument. The first is n, the dimension of the space in which the n-sphere is embedded. The second is a numpy array of shape (N, n) consisting of the n-dimensional unit vectors at which to center the distributions from which the samples are to be generated. The third is a numpy array of shape (N,) consisting of the scalar variance parameters of each distribution from which to generate samples. The fourth is a boolean flag that determines whether sampling should be done on the full surface of the n-sphere (if False) or on the hemisphere with reflecting boundary conditions for the diffusion kernel.

Example output from sample_spherical_kernel is:

>>> import numpy as np
>>> from spherediff.sample import sample_spherical_kernel
>>> np.random.seed(42)
>>> means = np.random.randn(5, 3)
>>> means /= np.linalg.norm(means, axis=1, keepdims=True)
>>> means
array([[ 0.60000205, -0.1670153 ,  0.78237039],
       [ 0.97717133, -0.15023209, -0.15022156],
       [ 0.86889694,  0.42224942, -0.25830898],
       [ 0.63675162, -0.5438697 , -0.54658314],
       [ 0.09351637, -0.73946664, -0.66666616]])
>>> vars = 0.1 * np.ones(5)
>>> sample_spherical_kernel(3, means, vars)
array([[ 0.30027556, -0.53104481,  0.79235472],
       [ 0.91657116, -0.39288942,  0.07439905],
       [ 0.81325411,  0.41495422, -0.40795926],
       [ 0.39907791, -0.44171124, -0.80350981],
       [ 0.16422958, -0.76019121, -0.62860001]])
>>> sample_spherical_kernel(3, means, vars, hemisphere=True)
array([[ 0.92723597,  0.02336567,  0.37374791],
       [ 0.99421791, -0.03878944, -0.10013055],
       [ 0.15771025,  0.6492883 , -0.74401087],
       [ 0.2418101 , -0.41127436, -0.87885225],
       [-0.11192408, -0.71437847, -0.69075061]])

The user may also compute the score function of the isotropic Gaussian distribution on the unit n-sphere, defined as the Riemannian gradient of the logarithm of the probability density. This may be done using the score_spherical_kernel function, which may be imported as follows:

from spherediff.score import score_spherical_kernel

This function takes four arguments and one additional, optional argument. The first is n, the dimension of the space in which the n-sphere is embedded. The second is a numpy array of shape (N, n) consisting of the n-dimensional unit vectors at which to evaluate the score function. The third is a numpy array of shape (N, n) consisting of the n-dimensional unit vectors at which to center the distributions of which the corresponding score functions will be evaluated. The fourth is a numpy array of shape (N,) consisting of the scalar variance parameters of each distribution. The fifth is a boolean flag that determines whether the distributions of interest are supported on the full surface of the n-sphere (if False) or on the hemisphere with reflecting boundary conditions for the diffusion kernel.

Example output from score_spherical_kernel is:

>>> import numpy as np
>>> from spherediff.sample import sample_spherical_kernel
>>> from spherediff.score import score_spherical_kernel
>>> np.random.seed(42)
>>> means = np.random.randn(5, 3)
>>> means /= np.linalg.norm(means, axis=1, keepdims=True)
>>> means
array([[ 0.60000205, -0.1670153 ,  0.78237039],
       [ 0.97717133, -0.15023209, -0.15022156],
       [ 0.86889694,  0.42224942, -0.25830898],
       [ 0.63675162, -0.5438697 , -0.54658314],
       [ 0.09351637, -0.73946664, -0.66666616]])
>>> vars = 0.1 * np.ones(5)
>>> x = sample_spherical_kernel(3, means, vars)
>>> x
array([[ 0.30027556, -0.53104481,  0.79235472],
       [ 0.91657116, -0.39288942,  0.07439905],
       [ 0.81325411,  0.41495422, -0.40795926],
       [ 0.39907791, -0.44171124, -0.80350981],
       [ 0.16422958, -0.76019121, -0.62860001]])
>>> score_spherical_kernel(3, x, means, vars)
array([[ 3.40175815,  3.11417869,  0.79800571],
       [ 1.12626049,  2.20918798, -2.20878198],
       [ 0.65201631,  0.12437106,  1.42627781],
       [ 2.65653386, -1.32241858,  2.04638588],
       [-0.69053884,  0.17827374, -0.39600546]])
>>> x = sample_spherical_kernel(3, means, vars, hemisphere=True)
>>> x
array([[ 0.92723597,  0.02336567,  0.37374791],
       [ 0.99421791, -0.03878944, -0.10013055],
       [ 0.15771025,  0.6492883 , -0.74401087],
       [ 0.2418101 , -0.41127436, -0.87885225],
       [-0.11192408, -0.71437847, -0.69075061]])
>>> score_spherical_kernel(3, x, means, vars, hemisphere=True)
array([[-1.90180502, -1.93688258,  4.83930095],
       [-0.09346005, -1.10146816, -0.50128883],
       [ 8.79701439,  0.34656156,  2.16716953],
       [ 4.43128597, -1.97152723,  2.14184844],
       [ 2.0106285 , -0.40206101,  0.09002668]])