# Directional Non-Commutative Monoidal Embeddings for MNIST

arXiv: https://arxiv.org/abs/2506.03472 · HTML: https://mlsanity.com/paper/2506.03472/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 4 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 2 of 2)

- [Directional Non-Commutative Monoidal Structures for Compositional Embeddings in Machine Learning](https://mlsanity.com/paper/2505.15507/) (2025)
- [Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators](https://mlsanity.com/paper/2505.24533/) (2025)

## Abstract (arXiv)

We present an empirical validation of the directional non-commutative monoidal embedding framework recently introduced in prior work~citeGodavarti2025monoidal. This framework defines learnable compositional embeddings using distinct non-commutative operators per dimension (axis) that satisfy an interchange law, generalizing classical one-dimensional transforms. Our primary goal is to verify that this framework can effectively model real data by applying it to a controlled, well-understood task: image classification on the MNIST dataset~citelecun1998gradient. A central hypothesis for why the proposed monoidal embedding works well is that it generalizes the Discrete Fourier Transform (DFT)~citeoppenheim1999discrete by learning task-specific frequency components instead of using fixed basis frequencies. We test this hypothesis by comparing learned monoidal embeddings against fixed DFT-based embeddings on MNIST. The results show that as the embedding dimensionality decreases (e.g., from 32 to 8 to 2), the performance gap between the learned monoidal embeddings and fixed DFT-based embeddings on MNIST grows increasingly large. This comparison is used as an analytic tool to explain why the framework performs well: the learnable embeddings can capture the most discriminative spectral components for the task. Overall, our experiments confirm that directional non-commutative monoidal embeddings are highly effective for representing image data, offering a compact learned representation that retains high task performance. The code used in this work is available at https://github.com/mahesh-godavarti/directionalcompositionmnist.