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REVIEW 3 major objections 6 minor 55 references

MFH: Marrying Frequency Domain with Handwritten Mathematical Expression Recognition

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A frequency-domain side stream improves handwritten math expression recognition across three baselines.

desk verdict First DCT-based frequency stream for HMER is a clean idea, but test-set tuning of hyperparameters likely inflates the reported gains. read the letter →

arxiv 2507.00430 v1 pith:2HTQCJZB submitted 2025-07-01 cs.CV

Huanxin Yang , Qiwen Wang This is my paper
classification cs.CV
keywords handwrittenmathematicalexpressionrecognitionfrequencydomainanalysisdiscretecosinetransformhigh-frequencyfeatureextractionencoder-decoderarchitecturefusionCROHMEbenchmark
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that handwritten mathematical expression recognition, which usually processes images only in the spatial domain, can be made more accurate by adding a parallel stream that operates on frequency-domain information. The proposed method, MFH, applies a discrete cosine transform to image patches, keeps only high-frequency coefficients, and fuses the resulting features with the spatial features of an existing encoder through a Fusion and Alignment Block. On the CROHME benchmarks, MFH improves three different baselines, including a 0.95-2.09 percentage-point gain over CoMER. The authors' central claim is that high-frequency components encode the contours and two-dimensional layout of formulas in a way that spatial features miss, and that this benefit grows for long formulas.

What carries the argument

The load-bearing object is a Patch-DCT pre-processing step combined with a Fusion and Alignment Block (FAB). Each $n\times n$ patch is transformed by the discrete cosine transform, and coefficients outside the bottom-right $m\times m$ high-frequency block are zeroed, producing an image that retains only high-frequency content. A small extractor of patch embedding, MLP layers, channel attention, and positional encoding turns this into a frequency feature $K$, which is aligned with the spatial feature $T$ through an attention map $A$: $K = A_1 K \otimes V_K$, $T = A_2 T \otimes V_T$, followed by direct addition. The FAB is what lets the two domains be combined without altering the backbone architecture.

What would settle it

Run MFH on CROHME with the complementary selection, keeping the top-left $5\times5$ low-frequency coefficients and zeroing the high frequencies. If that variant matches or beats MFH, the claim that high-frequency detail drives the gains is false. Re-tuning $m$ on a validation split only would also show whether the reported optimum is an artifact of test-set selection.

Watch

Extended reading notes

Core claim

The central claim is that feeding a DCT-derived, high-frequency-only feature stream into an HMER encoder, alongside the usual spatial stream, consistently improves expression-recognition accuracy. With patch size $n=8$ and retention size $m=5$, keeping only the bottom-right $5\times5$ DCT coefficients per patch and zeroing the rest, MFH-CoMER reaches 61.66%, 62.07%, and 63.72% expression recognition rate on CROHME 2014, 2016, and 2019, beating the reproduced CoMER baseline by 0.95, 2.09, and 1.50 percentage points respectively; MFH also improves the DWAP and ABM baselines. The paper attributes the gains to the frequency stream's ability to capture formula contours and layout, and shows the largest improvements on expressions of length 30 or more.

Load-bearing premise

The benefit of MFH rests on the premise that discarding all low-frequency DCT coefficients and keeping only the bottom-right $5\times5$ high-frequency block of each patch is a useful representation for recognizing structure, and the retention size $m$ is tuned on the CROHME test sets, so this particular choice is not independently validated.

Editorial extensions

If this is right

  • MFH is plug-and-play, so existing encoder-decoder HMER frameworks can adopt it without redesigning the decoder and still gain accuracy.
  • Because the gains concentrate in formulas of length 30 or more, the frequency stream specifically helps the complex two-dimensional layouts that are hardest for spatial-only recognizers.
  • DCT is preferable to FFT in this setup, indicating that real-valued cosine bases suit binary math-expression images better than complex Fourier coefficients.
  • The frequency stream remains beneficial when combined with data augmentation, so it can be layered onto stronger training pipelines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves decoder-side frequency attention untested; an editorial extension would be to feed frequency-aware cues into the attention decoder to attack the subscript and baseline errors shown in the attention maps.
  • Because the retention number $m$ is tuned on the test sets and produces a non-monotonic curve, a fairer test would tune $m$ on a validation split; if the gain shrinks, part of the reported improvement is a selection effect.
  • The same high-frequency fusion idea could transfer to other layout-dominated recognition tasks, such as chemical formula or table structure recognition, though the paper does not test this.
  • All experiments use CROHME, so whether the improvement generalizes to other handwriting styles and larger datasets remains an open testable question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes MFH, a plug-and-play frequency-domain stream for handwritten mathematical expression recognition (HMER). Input images are divided into n×n patches, transformed with the discrete cosine transform (DCT), and only the bottom-right m×m high-frequency coefficients are retained in each patch while other coefficients are zeroed. The resulting representation is passed through a small feature extractor consisting of patch embedding, MLP layers, channel attention, and positional encoding, and a Fusion and Alignment Block (FAB) combines this frequency feature with the spatial feature from the CNN encoder. On CROHME 2014/2016/2019, the authors report MFH-CoMER ExpRates of 61.66/62.07/63.72, outperforming their reproduced CoMER baseline by 0.95/2.09/1.50 points, and also report gains for DWAP and ABM baselines. Ablations analyze the contribution of the frequency input, DCT versus FFT, patch size, retention number m, and the components of the proposed block.

Significance. Frequency-domain structural cues are a plausible and relatively underexplored direction for HMER, and a plug-and-play module that consistently improves existing encoders would be a useful contribution. The paper releases source code and includes informative ablations, such as the DCT-versus-FFT comparison and the component analysis of the frequency extractor. However, the evaluation protocol selects hyperparameters and the final model checkpoint on the test sets themselves, so the reported improvements cannot currently be taken at face value; the central empirical claim still needs an unbiased evaluation with a proper validation split.

major comments (3)
  1. [Sec. 5.4, Fig. 5, Tabs. 4-6] The hyperparameters m and n are selected by maximizing the average ExpRate over the three CROHME test sets. Since m is chosen as the best of eight values evaluated on the same data that are later reported as results, the comparison between MFH and the baseline is confounded by selection on the test sets. The m-sweep itself illustrates the problem: m=3 yields 60.71 Avg. ExpRate, below the 61.00 baseline, while m=5 yields 62.54; picking the maximum of this non-monotonic curve on the test sets can create an apparent gain even if the frequency stream has no true benefit. Please re-run the experiments with a validation split disjoint from all reported test sets, or fix hyperparameters before seeing test data, report results for all swept values, and include variance across seeds. The same requirement applies to the component choices in Tabs. 4-6.
  2. [Sec. 5.2, Tab. 1] The paper states that CROHME 2014 is used as the validation set to select the best-performing model during training, and the same CROHME 2014 set is reported as a test set in Tab. 1. This makes the 2014 ExpRate and the averaged metrics used throughout Sec. 5.4 dependent on model selection and therefore not an independent measure of generalization. The authors should use a validation set that is disjoint from all three reported test sets, for example a held-out portion of the 8836-image training set or a different CROHME year, and state this protocol explicitly.
  3. [Sec. 4.1, Fig. 3] The preprocessing step is described as retaining the bottom-right m×m DCT coefficients and then reshaping the patches to the original image size, but it is not stated whether I′ is obtained by an inverse DCT, so that it is a spatial image with low frequencies suppressed, or by directly arranging the coefficient matrix as an image-like tensor. This distinction matters for interpreting the method as frequency-domain augmentation versus coefficient-input training, and it affects how the patch-embedding layer should be understood. Please clarify the exact operation.
minor comments (6)
  1. [Eq. (4)] In the positional encoding formula, the variable p appears inside the sine and cosine instead of the position x; the standard form should be sin(x/10000^(2i/d)) and cos(x/10000^(2i/d)).
  2. [Tabs. 1 and 2] Tab. 2 reports the CoMER baseline as 61.00, which is an average over the three test sets, while Tab. 1 lists per-set values; make the averaging convention explicit and consider reporting per-set numbers for all ablations.
  3. [Tab. 3] The FFT variant is only described as using FFT instead of DCT; specify how the complex FFT coefficients are converted to the real-valued input expected by the patch-embedding layer.
  4. [Fig. 6] The y-axis is labeled Avg. ExpRate, but the bars are grouped by formula-length interval; clarify that the average is taken over the three test sets for each length bin.
  5. [Sec. 5.5] The text says "We hypothesis in Sec. 1"; this should read "We hypothesize."
  6. [Sec. 5.1] Please include the training schedule, optimizer, learning rate, and number of epochs; the current implementation details give model dimensions and hardware but not these training hyperparameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MFH is an empirical architecture; reported gains are measured, not derived from the test targets.

full rationale

None of the paper's claimed derivation steps reduce to its own inputs by construction. The DCT transform and retained-coefficient selection are introduced as a preprocessing choice, not as a quantity fitted to the output labels and then renamed as a prediction. The central claim is an empirical comparison: MFH-CoMER improves over a reproduced CoMER baseline on the CROHME test sets. The hyperparameters m and n are selected using average ExpRate on the same test sets in Sec. 5.4, which is a genuine evaluation-protocol weakness that can inflate reported gains, but it is not a circular derivation: the final accuracy is still determined by training and inference on data, not by the selection rule alone. No self-citation is load-bearing: all cited works are external, and no uniqueness theorem or ansatz is imported from the authors' prior work. The FAB alignment equations and learnable vectors are architectural components whose effectiveness is measured by ablation, not asserted by definition. The limitations in Sec. 5.6 acknowledge unexplored directions but do not assert a missing proof or a circular step. Therefore, the paper's reasoning chain is self-contained with respect to the definition of circularity used here, and the score is 0.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim relies on a handful of architectural hyperparameters, some selected on the test sets, and on the untested assumption that high-frequency DCT coefficients are the useful part of the signal for HMER. No new physical or conceptual entities are introduced.

free parameters (5)
  • patch size n = 8
    Chosen from comparison of 8x8 vs 16x16 (Table 4) using test set average; not justified analytically.
  • retention number m = 5
    Selected by scanning m in Fig. 5 on the test sets' average ExpRate, which is fitting to test data.
  • output channel dimension C = 256
    Set in implementation, no ablation shown.
  • dropout rate in MLP layers = 0.3
    Set in implementation, no ablation shown.
  • number of MLP layers = 6
    Set for MFH-CoMER; no ablation shown.
assumptions (4)
  • standard math DCT is a linear transform with orthogonality properties that can be applied to image patches.
    Invoked in Sec 3.1 as the basis for Patch-DCT.
  • domain assumption High-frequency components encode structural and contour information of handwritten formulas.
    Central premise introduced in Sec 1 and used in Sec 4.1 to justify keeping only high-frequency coefficients.
  • domain assumption CNN backbone features and frequency features can be aligned additively after the FAB without harming recognition.
    The FAB design in Sec 4.3 assumes this merge is beneficial and trainable.
  • domain assumption The network remains trainable end-to-end with the added frequency stream.
    Assumed throughout Sec 4 and implied by the experimental setup.

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Cite this review

Pith. "Pith review of MFH: Marrying Frequency Domain with Handwritten Mathematical Expression Recognition." pith.science (2026). https://pith.science/paper/2HTQCJZB

@misc{pith2026250700430,
  author       = {Pith},
  title        = {Pith review of: MFH: Marrying Frequency Domain with Handwritten Mathematical Expression Recognition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HTQCJZB}},
  note         = {Machine review of arXiv:2507.00430}
}
read the original abstract

Handwritten mathematical expression recognition (HMER) suffers from complex formula structures and character layouts in sequence prediction. In this paper, we incorporate frequency domain analysis into HMER and propose a method that marries frequency domain with HMER (MFH), leveraging the discrete cosine transform (DCT). We emphasize the structural analysis assistance of frequency information for recognizing mathematical formulas. When implemented on various baseline models, our network exhibits a consistent performance enhancement, demonstrating the efficacy of frequency domain information. Experiments show that our MFH-CoMER achieves noteworthy accuracyrates of 61.66%/62.07%/63.72% on the CROHME 2014/2016/2019 test sets. The source code is available at https://github.com/Hryxyhe/MFH.

Figures

Figures reproduced from arXiv: 2507.00430 by the authors.

Figure 1
Figure 1. Comparison of different base￾lines and our method. To demonstrate the effectiveness of our MFH, we conduct experiments on the CROHME datasets. As shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The overview of MFH, which consists of a patch embedding layer, MLP [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The DCT pre-processing. The input image is first divided into patches, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The FAB used in the feature fusion stage. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Avg. ExpRate for different retention number [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Avg. ExpRate for different lengths of formulas. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Attention maps of CoMER [14] and our MFH-CoMER. recognition accuracy for long sequences, typically characterized by more complex spatial structures. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.