REVIEW 3 major objections 5 minor 4 references
Directional Non-Commutative Monoidal Embeddings for MNIST
T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper reports that learned monoidal embeddings outperform fixed truncated DFT features on MNIST, with the advantage growing as the embedding dimension falls, and attributes this to the model learning task-specific frequency components.
desk verdict A short empirical note undercut by a method description that can't compute its own embedding and a missing learned-linear baseline. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The core object is a pair of $d \times d$ orthogonal matrices $R_x$ and $R_y$, each a block-diagonal stacking of $2 \times 2$ rotation matrices $R(\theta) = [[\cos\theta, -\sin\theta], [\sin\theta, \cos\theta]]$, with independent angles per block per axis. Composing along an axis rotates the embedded content by powers of the axis matrix, so a whole image's embedding is a weighted sum of sinusoids with per-block horizontal frequency $\theta^x_k$ and vertical frequency $\theta^y_k$ (Eq. (3)). Because both matrices are block-diagonal in the same 2D subspaces, $R_x R_y = R_y R_x$, which satisfies the interchange law that makes row-then-column and column-then-row composition agree. In the experiments the angles are not fixed by formula but learned end-to-end by gradient descent, which is the mechanism that lets the embedding choose task-specific frequency components.
What would settle it
Look at the released code and verify whether the embedding for $d=32$ or $d=8$ is actually computed as the double sum over pixels of $p_{ij} R_y^i R_x^j e$ (Eq. (3)). If the code path instead projects the image by some other linear map, or if changing the order of the two matrix multiplications changes the output, then the central comparison in Table 1 does not test the monoidal embedding. A direct check is to evaluate both the sequential matrix-multiplication implementation and the explicit Eq. (3) sum on the same images and compare outputs exactly.
Extended reading notes
Core claim
The central claim is stated in Section 3.2: the directional monoidal embedding framework works well because it can adaptively learn frequency-like components tuned to the data and task. Concretely, for MNIST the learned embedding gives 96.5% accuracy at 32 dimensions versus 95.5% for the 32 lowest-frequency DFT features, 86.4% versus 75.3% at 8 dimensions, and 55.2% versus 21.0% at 2 dimensions. The authors argue that the fixed DFT features are stuck with low frequencies that may be non-discriminative, whereas the learned angles $\theta^x_k$ and $\theta^y_k$ concentrate the representation on the most class-informative oscillation patterns. At full 784 dimensions the monoidal embedding matches the linear-pixel baseline, which the paper treats as a sanity check that enough capacity loses no information.
Load-bearing premise
The load-bearing premise is that the implementation described in Section 3.1 indeed computes the monoidal embedding of Eq. (3); the text's matrix multiplications are dimensionally inconsistent as written, so if the code uses a different projection the experiment would not test the claimed framework.
Editorial extensions
If this is right
- At 32 dimensions, the learned monoidal embedding nearly matches a 128-unit MLP while using only tens of trainable parameters in $R_x$ and $R_y$, suggesting that compact spectral representations can preserve task-relevant information.
- The widening gap at 8 and 2 dimensions supports the interpretation that learned projections choose more discriminative features than fixed low-frequency Fourier coefficients.
- The full-dimensional case shows the framework loses no information when given enough capacity, matching a linear classifier on raw pixels.
- The monoidal embedding remains below the CNN baseline, so a single-layer linear spectral embedding is not yet competitive with learned nonlinear features on MNIST.
Reading between the lines
- One extension the paper leaves implicit is that the same learned-frequency mechanism should transfer to other grid-structured data, such as audio spectrograms or satellite image tiles, where compact discriminative spectral features matter.
- The result at 2 dimensions suggests that learned 2D sinusoidal projections can extract class structure that the DC and lowest fundamental DFT component lack; a direct test would be to visualize the learned angles and check which image patterns they align with.
- A natural extension the paper does not develop is adding a nonlinearity or multi-layer monoidal composition; on MNIST that path could be tested by whether a stacked model closes the gap to the CNN baseline.
- The commutativity constraint that gives the interchange law limits each block to the same 2D subspace for both axes; relaxing it with coupled blocks could permit anisotropic frequencies, at the cost of losing the exact interchange law.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents an empirical evaluation of the 'directional non-commutative monoidal embedding' framework on MNIST. The model embeds images via learnable block-diagonal rotation matrices for the x- and y-axes, followed by a linear classifier. The central empirical claim is that these learned embeddings outperform fixed low-frequency DFT features, with the gap widening as the embedding dimension shrinks (96.5 vs. 95.5 at d=32; 86.4 vs. 75.3 at d=8; 55.2 vs. 21.0 at d=2; Table 1). The authors interpret this as evidence that the framework works by adaptively learning task-specific frequency-like components. They also compare against an MLP and a CNN.
Significance. If the implementation is correct, the paper provides a useful sanity check of a theoretically motivated transform and demonstrates a compact learned representation that approaches MLP performance with far fewer parameters. The release of code, the explicit d=784 sanity check that matches the linear-classifier ceiling, and the controlled choice of MNIST are strengths. However, the significance is currently limited by an ambiguous and, as written, dimensionally inconsistent implementation description (Section 3.1), the absence of error bars or multiple seeds, and a DFT baseline that is a hand-picked low-frequency subset. These issues undermine confidence in the central frequency-learning interpretation.
major comments (3)
- [Section 3.1, Eq. (3)] The implementation described cannot compute Eq. (3) as written. The text states that the Ny × Nx image matrix is multiplied by R_x^T (d × d), which is only defined when Nx = d; for all reported d = 2, 8, 32 with Nx = 28 this multiplication is undefined. Even if one first embeds pixels into a d-dimensional space, the subsequent multiplication of the resulting Ny × d intermediate by Ry (d × d) produces an Ny × d matrix, not a d-dimensional embedding vector. The description also does not account for the powers R_y^i and R_x^j in Eq. (3). Please provide a precise tensor operation or pseudocode, or explicitly point to the released code, and demonstrate that the implementation matches Eq. (3). This is load-bearing because the low-dimensional results are the main evidence for the claim that the framework learns useful spectral components.
- [Section 3.2, Table 1] All accuracies are reported as single numbers without error bars or multiple seeds. The claim that the performance gap grows with reduced dimension relies in part on a small difference at d=32 (96.5 vs. 95.5). Without variance estimates or a significance test, the quantitative trend is not established. Please report the mean and standard deviation over at least several independent runs, or otherwise justify that the differences are not due to optimization noise.
- [Section 3.1, DFT baseline] The fixed DFT baseline uses only the 2, 8, or 32 lowest-frequency coefficients, which is an arbitrary projection and is expected to be suboptimal for classification. The observation that learned projections outperform this hand-picked baseline does not specifically support the monoidal structure; any learnable linear map could be expected to do better. To strengthen the interpretation in Section 3.2, add additional fixed baselines of the same dimensionality, such as PCA projections, random orthogonal projections, or a learned linear projection without the monoidal rotational constraint. This would help isolate the benefit attributable to the monoidal framework rather than to learnability alone.
minor comments (5)
- [Section 2.1, Eq. (2)] The derivation connecting the monoidal embedding to the DFT is informal; the assumption that each vt is 'proportional to some scalar feature' or 'structured similarly' is not stated precisely. A concrete one-dimensional construction would improve clarity.
- [Section 3.1] The description of the full DFT features is ambiguous: the text mentions both '784 real coefficients' and 'real and imaginary parts as separate features.' Please clarify the exact feature vector used in the full-dimensional DFT case.
- [Table 1] The d=784 monoidal embedding result is discussed in the text (about 97.5% accuracy) but is not included in Table 1; adding it would make the table complete and consistent with the narrative.
- [Throughout] The paper cites only the author's own prior work and standard signal processing textbooks. A brief comparison with other learned linear embeddings (e.g., PCA, learned spectrogram features) would help position the contribution in the broader literature.
- [Section 1 and 4] The phrase 'comparing favorably to standard MLP and CNN baselines' slightly overstates the results: at d=32 the monoidal model (96.5%) is below the MLP (97.2%) and well below the CNN (98.6%). Consider rewording to 'approaches the performance' as used elsewhere in the paper.
Circularity Check
No significant circularity: the MNIST experiment is an independent empirical comparison against fixed DFT features, and the cited prior framework is context, not evidence.
full rationale
The paper's only derived claim is empirical: learned rotation-parameterized embeddings outperform fixed truncated DFT features on MNIST, with the gap growing as dimension shrinks (Table 1). This is a self-contained experiment: Rx and Ry are trained on the MNIST training set and evaluated on the test set, and the DFT baseline is an external fixed transform. The theoretical framework from Refs. [1,2] is cited for background and for the block-diagonal rotation parameterization, but the paper re-derives the relevant formulas in Section 2 (Eqs. (1)-(3)) and does not rely on those citations to establish the empirical result. The interpretation that the model learns frequency-like components is a post hoc explanation of the gap, not a prediction derived from the framework. The Section 3.1 description of the implementation is dimensionally ambiguous relative to Eq. (3), but that is a correctness or reproducibility concern, not a circularity: nothing in the paper's argument reduces the empirical comparison to an assumed conclusion. No fitted parameter is renamed as a prediction, and no uniqueness claim from the author's prior work is invoked to force the framework. Under the review rules, this is a no-circularity finding.
Assumptions & free parameters
free parameters (3)
- Rotation angles theta_x_k, theta_y_k =
d parameters for embedding dimension d (e.g., 32, 8, 2) learned via Adam
- Linear classifier weights W, b =
d*10 + 10 parameters per model
- Pixel basis embedding vector e =
unspecified; text says 'fixed basis vector'
assumptions (4)
- domain assumption The monoidal embedding of a 2D image is given by the double summation in Eq. (3): E = sum_{i,j} p_ij R_y^i R_x^j e.
- ad hoc to paper The 32, 8, and 2 lowest-frequency DFT coefficients constitute an appropriate fixed spectral baseline.
- domain assumption Gradient descent on the cross-entropy loss finds good angles and classifier weights for all models.
- domain assumption MNIST is a representative controlled task for validating the framework.
Cite this review
Pith. "Pith review of Directional Non-Commutative Monoidal Embeddings for MNIST." pith.science (2026). https://pith.science/paper/7GYSD6FM
@misc{pith2026250603472,
author = {Pith},
title = {Pith review of: Directional Non-Commutative Monoidal Embeddings for MNIST},
year = {2026},
howpublished = {\url{https://pith.science/paper/7GYSD6FM}},
note = {Machine review of arXiv:2506.03472}
}
read the original abstract
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.
Reference graph
Works this paper leans on
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M. Godavarti, Directional Non-Commutative Monoidal Structures for Com- positional Embeddings in Machine Learning. arXiv:2505.15507, 2025
arXiv 2025
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[2]
Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators
M. Godavarti, Directional Non-Commutative Monoidal Structures with In- terchange Law via Commutative Generators. arXiv:2505.24533, 2025
work page Pith review arXiv 2025
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[3]
Alan V. Oppenheim and Ronald W. Schafer. Discrete-Time Signal Process- ing. Prentice Hall, 1999
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Yann LeCun, L´ eon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient- based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. 10
work page 1998
Reviewed August 7, 2026 · model on record in the stance chip above.
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