The Return of Structural Handwritten Mathematical Expression Recognition

Jakob Seitz; Radu Timofte; Tobias Lengfeld

REVIEW 4 major objections 5 minor 23 references

The Return of Structural Handwritten Mathematical Expression Recognition

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Structural math recognition returns, hitting 74.14% on CROHME-2023

desk verdict A genuinely useful dataset and a clean modular structural pipeline, but the auto-labeling loop makes the headline gains conditional on labeler consistency rather than independent ground truth. read the letter →

arxiv 2508.19773 v1 pith:5OL6LW2Q submitted 2025-08-27 cs.CV

Jakob Seitz , Tobias Lengfeld , Radu Timofte This is my paper

classification cs.CV

keywords HandwrittenMathematicalExpressionRecognitionStructuralStrokeLabelGraphAutomaticAnnotationGraph-BasedModelsExplainableAICROHME-2023DatasetAugmentation

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 structural, trace-to-symbol aligned handwriting recognition can compete with modern end-to-end systems if it is given enough high-quality structural training data. The authors build an automatic annotation network that converts existing LaTeX-labeled handwriting datasets into fully annotated ones—adding MathML relations and trace-group labels—and then train a modular five-stage pipeline on the enriched data. The result is 74.14% expression accuracy on CROHME-2023, competitive with encoder-decoder models, while every recognized symbol is linked back to its raw traces in a Stroke Label Graph. The paper also releases the annotated datasets, CROHME+ and MathWriting+, so the structural approach becomes reproducible and extendable.

What carries the argument

The Stroke Label Graph (SLG) is the central object: a directed graph whose nodes are trace groups (symbols) and whose edges are spatial relations (right, sup, sub, over, under, line_start), directly convertible to LaTeX. The argument is carried by two machines: (1) an automatic annotation network—a BiLSTM with fully connected layers that takes raw trace coordinates plus structural info extracted from the LaTeX label and outputs a binary segmentation mask per symbol—which produces the SLG training data; and (2) the five-stage modular recognition pipeline that independently optimizes segmentation, classification, and relation prediction, with a transformer correcting symbol classes using globa

What would settle it

Take a random sample of 1,000 auto-annotated expressions from the released CROHME+ dataset, have independent human annotators create trace-level MathML labels, and compare. If the disagreement rate is high for uncommon symbols or unusual writing styles, and if training the pipeline on only human-verified samples drops expression accuracy far below 74.14%, then the reported gains partly reflect labeler consistency rather than true structural understanding.

Watch

Extended reading notes

Core claim

The central discovery is that the perceived ceiling of structural HMER was largely a data problem, not an architecture problem. By training a neural network to align LaTeX labels with raw traces, the authors generate structural annotations (symbol segmentation, classification, and spatial relations) for hundreds of thousands of expressions that previously had only LaTeX labels. Trained on these auto-labeled annotations, a modular pipeline—graph-based trace sorting, dual-modal symbol classification, relation prediction, transformer-based correction, and a final relation refinement—achieves state-of-the-art results among structurally interpretable methods and narrows the gap to black-box encod

Load-bearing premise

The whole pipeline depends on the auto-generated structural labels being correct enough to serve as training targets, and the cross-checking filters use a classifier trained in the same label ecosystem rather than independent ground truth.

Editorial extensions

If this is right

If the central claim holds, structural HMER is no longer relegated to simple formulas: with enriched data, it can match or beat graph-based baselines on all five CROHME-2023 metrics (Seg, Sym, Rel, Exp, Stru).
Systems that need selective content updates or transparent error analysis—such as interactive math editors and grading tools—can use the SLG output to localize mistakes to specific strokes or symbols.
The released CROHME+ and MathWriting+ datasets give the community trace-level annotations for 374,000 expressions, removing the manual-labeling bottleneck that previously limited structural methods.
The auto-annotation network's cross-dataset generalization (trained on CROHME, labeling MathWriting with 254 classes) suggests that annotation transfer may work for other structured handwritten content.
Because all modules are trained independently, individual stages can be swapped or improved without retraining the whole system.

Reading between the lines

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

The paper's strongest implicit claim is that labeler consistency—not ground-truth correctness—is what drives the downstream gains; a direct comparison against human-verified labels would separate these two explanations.
If the auto-annotation approach transfers beyond math, the same LaTeX-to-trace alignment idea could provide structural supervision for handwritten chemical formulas, musical notation, or diagrams where a symbolic description already exists.
A testable extension: feeding the SLG structure (rather than only LaTeX text) into an LLM-based post-corrector could combine the interpretability of structural recognition with the fluency of modern language models.
The 1:1 trace-to-symbol constraint noted in the paper suggests that relaxing it—allowing shared or fragmented traces—is the natural next boundary; verifying the pipeline on cursive-style or multi-letter-symbol handwriting would show whether the approach generalizes beyond the CROHME symbol set.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take. The paper does two real things: it builds an auto-annotation network that turns LaTeX-plus-traces into MathML and trace-group labels at scale, and it releases CROHME+ and MathWriting+. Those datasets are the actual contribution; they let the community train structural recognizers without manual labeling. The modular five-stage pipeline is a sensible engineering effort, and the SLG output is a nice differentiator over encoder-decoder systems. The reported 74.14% on CROHME-2023 is competitive for a structural method, and the stage-by-stage ablations in Table 1 give a coherent story: generated annotations help segmentation, classification, and relations monotonically. The comparison against GGM-EGAT and G2G is fair and not cherry-picked.

The soft spot is the one the stress-test flagged. The auto-labeler is trained on CROHME ground truth, then used to label synthetic CROHME and MathWriting. The filters for those labels are classifiers trained in the same label ecosystem. For MathWriting, an expression is kept only if a reclassification matches the LaTeX label; for synthetic CROHME, top-10 match suffices. That means the retained data is biased toward what the labeler already recognizes. The reported 99.45% per-symbol accuracy is on CROHME validation with ground truth; there is no independent estimate of label quality on the newly generated data. So the gains from "with generated annotations" could partly reflect the pipeline learning to imitate the labeler's blind spots rather than doing true structural recognition. That concern is real but not fatal: the authors explicitly acknowledge the filtering may exclude rare patterns, and the released datasets let others test label quality directly.

Two smaller issues. The validation Exp of 82.77% versus the test/benchmark Exp of 74.14% is never reconciled; a reader needs a clear statement about which numbers come from where. There are also no error bars or multiple runs, and no code or weights as of writing. The paper is otherwise honest about limitations in Section 5—trace-sharing, domain scope, scalability—so I don't see any attempt to oversell.

Who is this for? Anyone working on HMER who wants interpretable outputs or a structurally annotated training set. It deserves a serious referee. Send it to review, and ask the authors to release code, reconcile the validation/test gap, and ideally measure labeler error on a small hand-annotated sample from MathWriting. If that check holds, the paper is a solid contribution.

Referee Report

4 major / 5 minor

Summary. The paper proposes an automatic annotation network that takes raw traces and a LaTeX label as input and predicts trace groups and spatial relations, generating MathML and Trace-Group annotations for CROHME synthetic data and MathWriting. Using these enriched CROHME+ labels, the authors train a modular structural recognition pipeline with five stages: trace segmentation, dual-modal symbol classification, primary relation prediction, transformer-based classification correction, and revised relation prediction. The system outputs a Stroke Label Graph (SLG) that links traces to symbols and is convertible to LaTeX. The central empirical claim is 74.14% expression accuracy on the CROHME-2023 benchmark, presented as competitive with encoder-decoder/LLM approaches while retaining full structural interpretability. The authors release CROHME+ and MathWriting+ datasets.

Significance. If validated, the work would be a notable result: it challenges the prevailing view that structural, modular HMER is inherently inferior to end-to-end encoder-decoder models, and it provides a scalable way to generate trace-level structural annotations without manual labeling. The explicitly interpretable SLG output is a practical advantage for error analysis and interactive applications, and the released datasets are potentially valuable community resources. The paper also gives a clear ablation story showing that synthetic structural annotations improve each stage of the pipeline. However, the central benefit depends on the correctness of auto-generated labels, and the current manuscript does not provide independent evidence of label quality for the majority of the newly annotated data. The reported Stage-wise versus final metrics are also internally inconsistent, and the validation/test numbers are presented without a clear split, so the headline accuracy is not yet fully supported.

major comments (4)

[Section 3.2 and 3.3] The filtering procedure is not independent of the label-generation process. For synthetic CROHME, a symbol is accepted if its class appears in the classifier's top-10 outputs; for MathWriting, an expression is retained only if reclassification matches the LaTeX label. Both classifiers are trained in the same label ecosystem as the auto-annotation network, so systematic labeler errors on uncommon symbols or unusual writing styles can be propagated rather than detected. The paper reports 99.45% per-symbol accuracy on CROHME validation, but the bulk of the new training data (synthetic CROHME and MathWriting) has no ground truth, and no accuracy estimate is given for the retained labels there. Since these labels are training targets for the downstream recognizer, the 74.14% test result could partly reflect learning labeler-consistent mappings rather than human-ground-truth structure. Please
[Table 1 vs. Table 2 and Section 4.2/4.3] The paper reports validation Exp = 82.77% in Table 1 and benchmark Exp = 74.14% in Table 2 without explicitly stating that one is validation and the other is test. That difference is plausible, but the manuscript should state the split clearly and discuss the generalization gap. More seriously, Table 1 lists Sym = 99.30% and Rel = 98.37% for the full system, while the text in Section 4.2 reports Sym = 96.73% after the correction module and Section 4.3 reports Rel = 96.98% for the primary relation stage. No explanation reconciles these numbers. Please provide a single end-to-end result table with each metric, the exact evaluation split, and the precise pipeline configuration, or explicitly state that the stage-by-stage numbers are before final integration.
[Section 3.1] The annotation network is described only at a high level: a BiLSTM plus fully connected layers predicts a binary segmentation mask per 'step', using reference symbols, next symbols, neighbours, and relations. The input representation is not specified in enough detail to reproduce the method: how are LaTeX labels tokenized, aligned to trace steps, and converted into the relation context? How is the network trained on CROHME-2023 when, as the paper notes, 92% of the training set lacks MathML and many trace-group labels are incorrect? The paper should specify the exact training subset, the input feature vector, and the decoding/merging procedure from per-step binary masks to final Trace Groups. This is load-bearing because the entire CROHME+ and MathWriting+ resources, and hence all downstream gains, depend on this component.
[Section 4.2 / 4.4] The classification module uses a dual-stream network with a BiLSTM pathway and a CNN pathway, and the correction transformer, yet the reported gain from adding the CNN is large (Sym 95.01% to 96.55%, Exp 64.08% to 72.60%) and from the correction module is small. However, the paper never reports whether the test-set 74.14% corresponds to the full system with corrected relations or to the 'classification correction' stage alone. The 'Integrated Pipeline' section implies all five stages are used, but the exact inference-time configuration for the Table 2 result is not stated. Please specify the exact inference configuration for the headline number and report all component metrics on the same test split.

minor comments (5)

[Section 3.3 / Section 5] The retention rate for MathWriting is stated as 66% in Section 3.3 and 67% in the same paragraph / Section 5. Please make the numbers consistent.
[References / Table 1] Table 1 cites 'G2G (2021) [4]', but reference [4] is an 'Edge-featured graph attention network' paper, not the G2G paper; the G2G paper appears as references [15] and [16] (duplicate). Please fix the citation mapping and remove the duplicate reference.
[Section 4.4] Typo: 'Raw tracs are partitioned' should be 'Raw traces are partitioned'.
[Figures 5 and 6] The example equations in Figures 5 and 6 are shown as raw LaTeX strings in the caption. If these are rendered outputs, it would help to show them as images so the reader can actually assess the segmentation and relation quality; otherwise, the figures are hard to interpret.
[General] No variance or multiple-run information is given for any of the reported numbers. Given that several claims rely on small differences (e.g., 0.14% gain from revised relation prediction), please report the number of runs and standard deviations, or at least state that all numbers are from a single training run.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the benchmark claim is measured on an external test set, and generated annotations are auxiliary training targets rather than the predicted outputs.

full rationale

The central claim (74.14% expression accuracy on CROHME-2023, Table 2) is an external benchmark result, not a fitted value. The auto-labeling system (Section 3) is trained on CROHME-2023 training data and generates trace-group/MathML annotations for synthetic CROHME and MathWriting; these annotations are used only as training supervision for the modular recognition system (Section 4). The recognizer is then evaluated on held-out CROHME test data whose ground-truth labels are independent of the annotation network. No equation defines the predicted label in terms of the input label; the LaTeX labels used by the annotation network are input information, not outputs being predicted. The cross-checking filters in Section 3.2 use a reclassification model that shares the CROHME label ecosystem, which is a real annotation-quality limitation (the paper itself notes the filters 'may exclude rare patterns'), but it does not make the downstream test accuracy true by construction. The paper contains no load-bearing self-citations and no imported uniqueness theorem. The difference between validation Exp (82.77%, Table 1) and benchmark Exp (74.14%, Table 2) reflects validation-versus-test reporting, not circularity. The derivation is therefore self-contained with respect to the benchmark claim.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper introduces no physical or mathematical entities; the main burden rests on data assumptions. The free parameters are standard deep-learning hyperparameters and dataset-construction thresholds. The most consequential assumption is that auto-generated labels are accurate enough to serve as ground truth for downstream training, especially for MathWriting where no independent structural labels exist.

free parameters (6)

Segmentation foreground loss weight w=5 = 5
Chosen in Section 4.1 to counteract class imbalance; no sensitivity analysis is reported.
MST nearest-neighbor count k=20 = 20
k=20 in the graph-based trace sorting; the paper reports 98.03% vs 96.89% for direct Euclidean sorting but fixed k is not ablated.
Augmentation hyperparameters (scaling sigma=0.2/0.4 clamped [0.2,5], shear 0.1, rotation 8 deg, translation 0.15) = as listed
Hand-set in Section 4.1 to simulate handwriting variation; no search or robustness analysis.
Classification BiLSTM hidden size h=256 and three layers = 256/3 layers
Section 4.2 architecture choice; no systematic tuning or ablation of hidden size is reported.
Correction transformer d_model=256, 8 heads, 3 layers = 256/8/3
Section 4.2 architecture choice for the global context correction module; depends on prior practice rather than demonstrated optimization.
Relation network hidden size 64 and 8 attention heads = 64/8
Section 4.3 architecture choice; the paper reports that removing multi-head attention has minor impact, so exact value is not critical.

assumptions (5)

domain assumption CROHME-2023 and MathWriting ground-truth LaTeX labels are correct and sufficiently complete for training and evaluation.
Used throughout the paper as the source of truth for both annotation generation and benchmark evaluation; no manual verification is described.
domain assumption The auto-labeler's validation accuracy on CROHME transfers to synthetic CROHME data and to MathWriting, which has no structural ground truth.
Section 3.3 reports 84% and 66-67% annotation success on those corpora, but MathWriting labels are only checked by reclassification, not by independent ground truth.
domain assumption All symbols in the target domain consist of disjoint trace groups with a 1:1 trace-to-symbol mapping.
Stated as a limitation in Section 5 ('Trace-Sharing Constraint'); the segmentation, graph sorting, and tree generation stages all depend on this assumption.
domain assumption The relation set R = {right, sup, sub, over, under, line_start} is sufficient to express the structural content of CROHME-2023 expressions.
Defined in Section 4.3; the paper lists matrices, overlines, and additional symbols as future work, confirming the limitation.
domain assumption Processing (x,y) trace coordinates without timestamps preserves enough temporal structure for the BiLSTM models.
Section 3.1 notes that most CROHME samples lack timestamps, so only spatial coordinates are used and are treated as a sequential signal.

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Cite this review

Pith. "Pith review of The Return of Structural Handwritten Mathematical Expression Recognition." pith.science (2026). https://pith.science/paper/5OL6LW2Q

@misc{pith2026250819773,
  author       = {Pith},
  title        = {Pith review of: The Return of Structural Handwritten Mathematical Expression Recognition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5OL6LW2Q}},
  note         = {Machine review of arXiv:2508.19773}
}

read the original abstract

Handwritten Mathematical Expression Recognition is foundational for educational technologies, enabling applications like digital note-taking and automated grading. While modern encoder-decoder architectures with large language models excel at LaTeX generation, they lack explicit symbol-to-trace alignment, a critical limitation for error analysis, interpretability, and spatially aware interactive applications requiring selective content updates. This paper introduces a structural recognition approach with two innovations: 1 an automatic annotation system that uses a neural network to map LaTeX equations to raw traces, automatically generating annotations for symbol segmentation, classification, and spatial relations, and 2 a modular structural recognition system that independently optimizes segmentation, classification, and relation prediction. By leveraging a dataset enriched with structural annotations from our auto-labeling system, the proposed recognition system combines graph-based trace sorting, a hybrid convolutional-recurrent network, and transformer-based correction to achieve competitive performance on the CROHME-2023 benchmark. Crucially, our structural recognition system generates a complete graph structure that directly links handwritten traces to predicted symbols, enabling transparent error analysis and interpretable outputs.

Figures

Figures reproduced from arXiv: 2508.19773 by the authors.

**Figure 3.** Annotated and total expression counts for CROHME and MathWriting [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** System overview with five stages: (1) trace segmentation, (2) dual-modal [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: correctly predicted equations (a) A = R dxh (x) P j Bj (x) bj (x) (b) F (x, y) = sin xey Our system achieves 74.14% expression accuracy on CROHME-2023 while maintaining full structural interpretability through SLG generation. While this [PITH_FULL_IMAGE:figures/full_f…

**Figure 6.** Figure 6: highlights remaining challenges in distinguishing visually similar subexpressions [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

Discussion (0). Sign in to comment.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Syntax-directed recognition of hand-printed two-dimensional mathematics

Robert H Anderson. Syntax-directed recognition of hand-printed two-dimensional mathematics. InSymposium on interactive systems for experimental applied math- ematics: Proceedings of the Association for Computing Machinery Inc. Symposium, pages 436–459, 1967

work page 1967

[2] [2]

A syntactic approach for handwritten mathematical formula recognition

Abdelwaheb Belaid and Jean-Paul Haton. A syntactic approach for handwritten mathematical formula recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, (1):105–111, 1984

work page 1984

[3] [3]

A method for the structural analysis of two-dimensional mathe- matical expressions

Shi-Kuo Chang. A method for the structural analysis of two-dimensional mathe- matical expressions. information sciences, 2(3):253–272, 1970

work page 1970

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Jun Chen and Haopeng Chen. Edge-featured graph attention network. arXiv preprint arXiv:2101.07671, 2021

arXiv 2021

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Mathwriting: A dataset for handwritten mathematical expression recognition

Philippe Gervais, Asya Fadeeva, and Andrii Maksai. Mathwriting: A dataset for handwritten mathematical expression recognition. arXiv preprint arXiv:2404.10690, 2024

arXiv 2024

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