# Learning Holographic Reduced Representations with Clifford Variational Autoencoders

arXiv: https://arxiv.org/abs/2609.28409 · HTML: https://mlsanity.com/paper/2609.28409/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [On the Binding Problem in Artificial Neural Networks](https://mlsanity.com/paper/2012.05208/) (2020)
- [Cyclical Annealing Schedule: A Simple Approach to Mitigating KL Vanishing](https://mlsanity.com/paper/1903.10145/) (2019)
- [Holographic Embeddings of Knowledge Graphs](https://mlsanity.com/paper/1510.04935/) (2015)
- [Spherical Latent Spaces for Stable Variational Autoencoders](https://mlsanity.com/paper/1808.10805/) (2018)
- [A comparison of Vector Symbolic Architectures](https://mlsanity.com/paper/2001.11797/) (2020)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [The Power Spherical distribution](https://mlsanity.com/paper/2006.04437/) (2020)
- [Learning with Holographic Reduced Representations](https://mlsanity.com/paper/2109.02157/) (2021)
- [Functional Regularization for Reinforcement Learning via Learned Fourier Features](https://mlsanity.com/paper/2112.03257/) (2021)
- [Hyperspherical Latents Improve Continuous-Token Autoregressive Generation](https://mlsanity.com/paper/2509.24335/) (2025)
- [Recasting Self-Attention with Holographic Reduced Representations](https://mlsanity.com/paper/2305.19534/) (2023)
- [On PyTorch Implementation of Density Estimators for von Mises-Fisher and Its Mixture](https://mlsanity.com/paper/2102.05340/) (2021)
- [Audio Fingerprinting with Holographic Reduced Representations](https://mlsanity.com/paper/2406.13139/) (2024)
- [Multi-label Learning with Random Circular Vectors](https://mlsanity.com/paper/2407.05656/) (2024)

## Abstract (arXiv)

Vector Symbolic Algebras project data structures into a hyperdimensional vector space through the application of their vector algebras to randomly generated atomic vector symbols and fractional power encodings of real-valued data. Embedding unstructured data remains an open question. We present textitClifford-VAE, a variational autoencoder that learns to project data onto a Clifford torus in arbitrary dimensions. Experiments using the MNIST, FashionMNIST, and CIFAR-10 datasets demonstrate that Clifford-VAE produces representations that are competitive with those produced by Gaussian and Hyperspherical VAEs for semi-supervised classification tasks while outperforming Gaussian and Hyperspherical counterparts in the VSA benchmark tests of self-binding and unbinding, role-filler recovery, and bundle capacity. Clifford-VAE provides a principled technique for grounding perceptual data into a symbolic reasoning framework, providing a new approach to a long-standing problem in the VSA literature.