Handwritten Mathematical Expression Recognition (HMER) remains a persistent challenge in Optical Character Recognition (OCR) due to the inherent freedom of symbol layout and variability in handwriting styles. Prior methods have faced performance bottlenecks, proposing isolated architectural modifications that are difficult to integrate coherently into a unified framework. Meanwhile, recent advances in pretrained vision-language models (VLMs) have demonstrated strong cross-task generalization, offering a promising foundation for developing unified solutions. In this paper, we introduce Uni-MuMER, which fully fine-tunes a VLM for the HMER task without modifying its architecture, effectively injecting domain-specific knowledge into a generalist framework. Our method integrates three data-driven tasks: Tree-Aware Chain-of-Thought (Tree-CoT) for structured spatial reasoning, Error-Driven Learning (EDL) for reducing confusion among visually similar characters, and Symbol Counting (SC) for improving recognition consistency in long expressions. Experiments on the CROHME and HME100K datasets show that Uni-MuMER achieves new state-of-the-art performance, surpassing the best lightweight specialized model SSAN by 16.31% and the top-performing VLM Gemini2.5-flash by 24.42% in the zero-shot setting. Our datasets, models, and code are open-sourced at: https://github.com/BFlameSwift/Uni-MuMER
Handwritten Mathematical Expression Recognition (HMER) has extensive applications in automated grading and office automation. However, existing sequence-based decoding methods, which directly predict $\LaTeX$ sequences, struggle to understand and model the inherent tree structure of $\LaTeX$ and often fail to ensure syntactic correctness in the decoded results. To address these challenges, we propose a novel model named TAMER (Tree-Aware Transformer) for handwritten mathematical expression recognition. TAMER introduces an innovative Tree-aware Module while maintaining the flexibility and efficient training of Transformer. TAMER combines the advantages of both sequence decoding and tree decoding models by jointly optimizing sequence prediction and tree structure prediction tasks, which enhances the model's understanding and generalization of complex mathematical expression structures. During inference, TAMER employs a Tree Structure Prediction Scoring Mechanism to improve the structural validity of the generated $\LaTeX$ sequences. Experimental results on CROHME datasets demonstrate that TAMER outperforms traditional sequence decoding and tree decoding models, especially in handling complex mathematical structures, achieving state-of-the-art (SOTA) performance.
Recently, Handwritten Mathematical Expression Recognition (HMER) has gained considerable attention in pattern recognition for its diverse applications in document understanding. Current methods typically approach HMER as an image-to-sequence generation task within an autoregressive (AR) encoder-decoder framework. However, these approaches suffer from several drawbacks: 1) a lack of overall language context, limiting information utilization beyond the current decoding step; 2) error accumulation during AR decoding; and 3) slow decoding speed. To tackle these problems, this paper makes the first attempt to build a novel bottom-up Non-AutoRegressive Modeling approach for HMER, called NAMER. NAMER comprises a Visual Aware Tokenizer (VAT) and a Parallel Graph Decoder (PGD). Initially, the VAT tokenizes visible symbols and local relations at a coarse level. Subsequently, the PGD refines all tokens and establishes connectivities in parallel, leveraging comprehensive visual and linguistic contexts. Experiments on CROHME 2014/2016/2019 and HME100K datasets demonstrate that NAMER not only outperforms the current state-of-the-art (SOTA) methods on ExpRate by 1.93%/2.35%/1.49%/0.62%, but also achieves significant speedups of 13.7x and 6.7x faster in decoding time and overall FPS, proving the effectiveness and efficiency of NAMER.
Handwritten Mathematical Expression Recognition (HMER) has wide applications in human-machine interaction scenarios, such as digitized education and automated offices. Recently, sequence-based models with encoder-decoder architectures have been commonly adopted to address this task by directly predicting LaTeX sequences of expression images. However, these methods only implicitly learn the syntax rules provided by LaTeX, which may fail to describe the position and hierarchical relationship between symbols due to complex structural relations and diverse handwriting styles. To overcome this challenge, we propose a position forest transformer (PosFormer) for HMER, which jointly optimizes two tasks: expression recognition and position recognition, to explicitly enable position-aware symbol feature representation learning. Specifically, we first design a position forest that models the mathematical expression as a forest structure and parses the relative position relationships between symbols. Without requiring extra annotations, each symbol is assigned a position identifier in the forest to denote its relative spatial position. Second, we propose an implicit attention correction module to accurately capture attention for HMER in the sequence-based decoder architecture. Extensive experiments validate the superiority of PosFormer, which consistently outperforms the state-of-the-art methods 2.03%/1.22%/2.00%, 1.83%, and 4.62% gains on the single-line CROHME 2014/2016/2019, multi-line M2E, and complex MNE datasets, respectively, with no additional latency or computational cost. Code is available at https://github.com/SJTU-DeepVisionLab/PosFormer.
Significant progress has been made in the field of handwritten mathematical expression recognition, while existing encoder-decoder methods are usually difficult to model global information in $LaTeX$. Therefore, this paper introduces a novel approach, Implicit Character-Aided Learning (ICAL), to mine the global expression information and enhance handwritten mathematical expression recognition. Specifically, we propose the Implicit Character Construction Module (ICCM) to predict implicit character sequences and use a Fusion Module to merge the outputs of the ICCM and the decoder, thereby producing corrected predictions. By modeling and utilizing implicit character information, ICAL achieves a more accurate and context-aware interpretation of handwritten mathematical expressions. Experimental results demonstrate that ICAL notably surpasses the state-of-the-art(SOTA) models, improving the expression recognition rate (ExpRate) by 2.25\%/1.81\%/1.39\% on the CROHME 2014/2016/2019 datasets respectively, and achieves a remarkable 69.06\% on the challenging HME100k test set. We make our code available on the GitHub: https://github.com/qingzhenduyu/ICAL
Recently, most handwritten mathematical expression recognition (HMER) methods adopt the encoder-decoder networks, which directly predict the markup sequences from formula images with the attention mechanism. However, such methods may fail to accurately read formulas with complicated structure or generate long markup sequences, as the attention results are often inaccurate due to the large variance of writing styles or spatial layouts. To alleviate this problem, we propose an unconventional network for HMER named Counting-Aware Network (CAN), which jointly optimizes two tasks: HMER and symbol counting. Specifically, we design a weakly-supervised counting module that can predict the number of each symbol class without the symbol-level position annotations, and then plug it into a typical attention-based encoder-decoder model for HMER. Experiments on the benchmark datasets for HMER validate that both joint optimization and counting results are beneficial for correcting the prediction errors of encoder-decoder models, and CAN consistently outperforms the state-of-the-art methods. In particular, compared with an encoder-decoder model for HMER, the extra time cost caused by the proposed counting module is marginal. The source code is available at https://github.com/LBH1024/CAN.
The Transformer-based encoder-decoder architecture has recently made significant advances in recognizing handwritten mathematical expressions. However, the transformer model still suffers from the lack of coverage problem, making its expression recognition rate (ExpRate) inferior to its RNN counterpart. Coverage information, which records the alignment information of the past steps, has proven effective in the RNN models. In this paper, we propose CoMER, a model that adopts the coverage information in the transformer decoder. Specifically, we propose a novel Attention Refinement Module (ARM) to refine the attention weights with past alignment information without hurting its parallelism. Furthermore, we take coverage information to the extreme by proposing self-coverage and cross-coverage, which utilize the past alignment information from the current and previous layers. Experiments show that CoMER improves the ExpRate by 0.61%/2.09%/1.59% compared to the current state-of-the-art model, and reaches 59.33%/59.81%/62.97% on the CROHME 2014/2016/2019 test sets.
The step function is one of the simplest and most natural activation functions for deep neural networks (DNNs). As it counts 1 for positive variables and 0 for others, its intrinsic characteristics (e.g., discontinuity and no viable information of subgradients) impede its development for several decades. Even if there is an impressive body of work on designing DNNs with continuous activation functions that can be deemed as surrogates of the step function, it is still in the possession of some advantageous properties, such as complete robustness to outliers and being capable of attaining the best learning-theoretic guarantee of predictive accuracy. Hence, in this paper, we aim to train DNNs with the step function used as an activation function (dubbed as 0/1 DNNs). We first reformulate 0/1 DNNs as an unconstrained optimization problem and then solve it by a block coordinate descend (BCD) method. Moreover, we acquire closed-form solutions for sub-problems of BCD as well as its convergence properties. Furthermore, we also integrate $\ell_{2,0}$-regularization into 0/1 DNN to accelerate the training process and compress the network scale. As a result, the proposed algorithm has a high performance on classifying MNIST and Fashion-MNIST datasets. As a result, the proposed algorithm has a desirable performance on classifying MNIST, FashionMNIST, Cifar10, and Cifar100 datasets.
Handwritten mathematical expression recognition (HMER) is a challenging task that has many potential applications. Recent methods for HMER have achieved outstanding performance with an encoder-decoder architecture. However, these methods adhere to the paradigm that the prediction is made "from one character to another", which inevitably yields prediction errors due to the complicated structures of mathematical expressions or crabbed handwritings. In this paper, we propose a simple and efficient method for HMER, which is the first to incorporate syntax information into an encoder-decoder network. Specifically, we present a set of grammar rules for converting the LaTeX markup sequence of each expression into a parsing tree; then, we model the markup sequence prediction as a tree traverse process with a deep neural network. In this way, the proposed method can effectively describe the syntax context of expressions, alleviating the structure prediction errors of HMER. Experiments on three benchmark datasets demonstrate that our method achieves better recognition performance than prior arts. To further validate the effectiveness of our method, we create a large-scale dataset consisting of 100k handwritten mathematical expression images acquired from ten thousand writers. The source code, new dataset, and pre-trained models of this work will be publicly available.
Handwritten mathematical expression recognition aims to automatically generate LaTeX sequences from given images. Currently, attention-based encoder-decoder models are widely used in this task. They typically generate target sequences in a left-to-right (L2R) manner, leaving the right-to-left (R2L) contexts unexploited. In this paper, we propose an Attention aggregation based Bi-directional Mutual learning Network (ABM) which consists of one shared encoder and two parallel inverse decoders (L2R and R2L). The two decoders are enhanced via mutual distillation, which involves one-to-one knowledge transfer at each training step, making full use of the complementary information from two inverse directions. Moreover, in order to deal with mathematical symbols in diverse scales, an Attention Aggregation Module (AAM) is proposed to effectively integrate multi-scale coverage attentions. Notably, in the inference phase, given that the model already learns knowledge from two inverse directions, we only use the L2R branch for inference, keeping the original parameter size and inference speed. Extensive experiments demonstrate that our proposed approach achieves the recognition accuracy of 56.85 % on CROHME 2014, 52.92 % on CROHME 2016, and 53.96 % on CROHME 2019 without data augmentation and model ensembling, substantially outperforming the state-of-the-art methods. The source code is available in https://github.com/XH-B/ABM.
Encoder-decoder models have made great progress on handwritten mathematical expression recognition recently. However, it is still a challenge for existing methods to assign attention to image features accurately. Moreover, those encoder-decoder models usually adopt RNN-based models in their decoder part, which makes them inefficient in processing long $\LaTeX{}$ sequences. In this paper, a transformer-based decoder is employed to replace RNN-based ones, which makes the whole model architecture very concise. Furthermore, a novel training strategy is introduced to fully exploit the potential of the transformer in bidirectional language modeling. Compared to several methods that do not use data augmentation, experiments demonstrate that our model improves the ExpRate of current state-of-the-art methods on CROHME 2014 by 2.23%. Similarly, on CROHME 2016 and CROHME 2019, we improve the ExpRate by 1.92% and 2.28% respectively.
Handwritten mathematical expression recognition is a challenging problem due to the complicated two-dimensional structures, ambiguous handwriting input and variant scales of handwritten math symbols. To settle this problem, we utilize the attention based encoder-decoder model that recognizes mathematical expression images from two-dimensional layouts to one-dimensional LaTeX strings. We improve the encoder by employing densely connected convolutional networks as they can strengthen feature extraction and facilitate gradient propagation especially on a small training set. We also present a novel multi-scale attention model which is employed to deal with the recognition of math symbols in different scales and save the fine-grained details that will be dropped by pooling operations. Validated on the CROHME competition task, the proposed method significantly outperforms the state-of-the-art methods with an expression recognition accuracy of 52.8% on CROHME 2014 and 50.1% on CROHME 2016, by only using the official training dataset.
In this paper we investigate image classification with computational resource limits at test time. Two such settings are: 1. anytime classification, where the network's prediction for a test example is progressively updated, facilitating the output of a prediction at any time; and 2. budgeted batch classification, where a fixed amount of computation is available to classify a set of examples that can be spent unevenly across "easier" and "harder" inputs. In contrast to most prior work, such as the popular Viola and Jones algorithm, our approach is based on convolutional neural networks. We train multiple classifiers with varying resource demands, which we adaptively apply during test time. To maximally re-use computation between the classifiers, we incorporate them as early-exits into a single deep convolutional neural network and inter-connect them with dense connectivity. To facilitate high quality classification early on, we use a two-dimensional multi-scale network architecture that maintains coarse and fine level features all-throughout the network. Experiments on three image-classification tasks demonstrate that our framework substantially improves the existing state-of-the-art in both settings.