Spherical Harmonics

This package implements spherical harmonics in d-dimensions in Python. The spherical harmonics are defined as zonal functions through the Gegenbauer polynomials and a fundamental system of points (see Dai and Xu (2013), defintion 3.1). The spherical harmonics form a ortho-normal set on the hypersphere. This package implements a greedy algorithm to compute the fundamental set for dimensions up to 20.

The computations of this package can be carried out in either TensorFlow, Pytorch, Jax or NumPy. A specific backend can be chosen by simply importing it as follows

import spherical_harmonics.tensorflow  # noqa

Example

3 Dimensional

import tensorflow as tf
import spherical_harmonics.tensorflow  # run computation in TensorFlow

from spherical_harmonics import SphericalHarmonics
from spherical_harmonics.utils import l2norm

dimension = 3
max_degree = 10
# Returns all the spherical harmonics in dimension 3 up to degree 10.
Phi = SphericalHarmonics(dimension, max_degree)

x = tf.random.normal((101, dimension))  # Create random points to evaluate Phi
x = x / tf.norm(x, axis=1, keepdims=True)  # Normalize vectors
out = Phi(x)  # Evaluate spherical harmonics at `x`

# In 3D there are (2 * degree + 1) spherical harmonics per degree,
# so in total we have 400 spherical harmonics of degree 20 and smaller.
num_harmonics = 0
for degree in range(max_degree):
    num_harmonics += 2 * degree + 1
assert num_harmonics == 100

assert out.shape == (101, num_harmonics)

4 Dimensional

The setup for 4 dimensional spherical harmonics is very similar to the 3D case. Note that there are more spherical harmonics now of degree smaller than 20.

import numpy as np
from spherical_harmonics import SphericalHarmonics
from spherical_harmonics.utils import l2norm

dimension = 4
max_degree = 10
# Returns all the spherical harmonics of degree 4 up to degree 10.
Phi = SphericalHarmonics(dimension, max_degree)

x = np.random.randn(101, dimension)  # Create random points to evaluation Phi
x = x / l2norm(x)  # normalize vectors
out = Phi(x)  # Evaluate spherical harmonics at `x`

# In 4D there are (degree + 1)**2 spherical harmonics per degree,
# so in total we have 385 spherical harmonics of degree 20 and smaller.
num_harmonics = 0
for degree in range(max_degree):
    num_harmonics += (degree + 1) ** 2
assert num_harmonics == 385

assert out.shape == (101, num_harmonics)

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NOTE

The fundamental systems up to dimensions 20 are precomputed and stored in spherical_harmonics/fundamental_system. For each dimension we precompute the first amount of spherical harmonics. This means that in each dimension we support a varying number of maximum degree (max_degree) and number of spherical harmonics:

Dimension	Max Degree	Number Harmonics
3	34	1156
4	20	2870
5	10	16170
6	8	1254
7	7	1386
8	6	1122
9	6	1782
10	6	2717
11	5	1287
12	5	1729
13	5	2275
14	5	2940
15	5	3740
16	4	952
17	4	1122
18	4	1311
19	4	1520
20	4	1750

To precompute a larger fundamental system for a dimension run the following script

cd spherical_harmonics
python fundament_set.py

after specifying the desired options in the file.

---

Installation

The package is now available on PyPI under the name of spherical-harmonics-basis.

Simply run

pip install spherical-harmonics-basis

Citation

If this code was useful for your research, please consider citing the following paper:

@inproceedings{Dutordoir2020spherical,
  title     = {{Sparse Gaussian Processes with Spherical Harmonic Features}},
  author    = {Dutordoir, Vincent and Durrande, Nicolas and Hensman, James},
  booktitle = {Proceedings of the 37th International Conference on Machine Learning (ICML)},
  date      = {2020},
}