clifford-torus 0.1.1

PyTorch distributions on the Clifford torus (S^1)^d for HRR/VSA-compatible latent spaces

Clifford Torus distribution

A PyTorch implementation of distributions on the Clifford torus (S^1)^d, as used by Clifford-VAE [1] to learn Holographic Reduced Representation (HRR) / Vector Symbolic Algebra-compatible latent spaces.

This distribution samples points on the Clifford torus (S^1)^d ⊂ R^(2d). This package also includes the PowerSpherical and HypersphericalUniform distributions (adapted from nicola-decao/power_spherical [2]), since the Clifford torus's per-circle concentration can be parameterized with either a von Mises or a Power Spherical distribution.

Dependencies

• python >= 3.9
• torch >= 1.10

Installation

pip install clifford-torus

or from source:

git clone https://github.com/momalekabid/clifford-torus
cd clifford-torus
pip install .

Structure

• clifford_torus/distributions.py: PowerSpherical, HypersphericalUniform, CliffordTorusUniform, CliffordTorusDistribution (von Mises concentration), CliffordPowerSphericalDistribution (Power Spherical concentration).

Usage

Differentiable sampling on a d-dimensional Clifford torus, returned as a real vector of length 2d whose non-DC Fourier coefficients all have unit magnitude:

import torch
from clifford_torus import CliffordPowerSphericalDistribution, CliffordTorusUniform

d = 8
loc = torch.zeros(d, requires_grad=True)          # per-circle mean angle
concentration = torch.full((d,), 4.0, requires_grad=True)

q = CliffordPowerSphericalDistribution(loc, concentration)
z = q.rsample()               # shape (2*d,), unit-magnitude fourier coefficients
z.sum().backward()

KL divergence against the uniform prior on the torus:

p = CliffordTorusUniform(dim=d)
torch.distributions.kl_divergence(q, p)

z from a CliffordPowerSphericalDistribution/CliffordTorusDistribution is directly usable in an HRR/VSA: bind two codes with circular convolution (torch.fft.ifft(torch.fft.fft(a) * torch.fft.fft(b)).real), and unbind with the exact inverse since every Fourier coefficient has unit magnitude.

references

@article{abid2026clifford,
  title={Learning Holographic Reduced Representations with Clifford Variational Autoencoders},
  author={Abid, Mohamed Malek and Furlong, P. Michael},
  year={2026}
}

for the underlying PowerSpherical distribution we use:

@article{decao2020power,
  title={The Power Spherical distribution},
  author={De Cao, Nicola and Aziz, Wilker},
  journal={Proceedings of the 37th International Conference on Machine Learning, INNF+},
  year={2020}
}

Some optional memory optimizations implemented following this post.

Visualization

Stereographic projection of spheres centered at points on the Clifford torus inside the 3-sphere, rotated in the zw-plane and projected down to R^3. From Clayton Shonkwiler's Wolfram Community post, "[GIF] Inside (Stereographic projection of points on the Clifford torus)".

@misc{shonkwiler2017inside,
  title={[GIF] Inside (Stereographic projection of points on the Clifford torus)},
  author={Shonkwiler, Clayton},
  howpublished={Wolfram Community},
  year={2017},
  url={https://community.wolfram.com/groups/-/m/t/1245707}
}

License

MIT