Directional Non-Commutative Monoidal Embeddings for MNIST
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908392567603200 |
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| author | Godavarti, Mahesh |
| author_facet | Godavarti, Mahesh |
| contents | We present an empirical validation of the directional non-commutative monoidal embedding framework recently introduced in prior work~\cite{Godavarti2025monoidal}. 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~\cite{lecun1998gradient}. A central hypothesis for why the proposed monoidal embedding works well is that it generalizes the Discrete Fourier Transform (DFT)~\cite{oppenheim1999discrete} 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/directional_composition_mnist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Directional Non-Commutative Monoidal Embeddings for MNIST Godavarti, Mahesh Machine Learning 20-XX, 08A02 F.4.1; I.2 We present an empirical validation of the directional non-commutative monoidal embedding framework recently introduced in prior work~\cite{Godavarti2025monoidal}. 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~\cite{lecun1998gradient}. A central hypothesis for why the proposed monoidal embedding works well is that it generalizes the Discrete Fourier Transform (DFT)~\cite{oppenheim1999discrete} 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/directional_composition_mnist. |
| title | Directional Non-Commutative Monoidal Embeddings for MNIST |
| topic | Machine Learning 20-XX, 08A02 F.4.1; I.2 |
| url | https://arxiv.org/abs/2506.03472 |