GryphOne: Symbol-Aware Masked Diffusion for Structural Refinement in Offline Handwritten Mathematical Expression Recognition

Handwritten Mathematical Expression Generation (HMEG) is challenging due to the complex two-dimensional layouts and long-range structural dependencies of mathematical expressions. Existing methods typically rely on explicit spatial supervision, such as symbol-level bounding boxes, which incurs high annotation costs and limits scalability. In this work, we propose DiffMath, a symbol- and graph-aware latent diffusion framework that leverages the hierarchical structure inherent in LaTeX as a structural prior, eliminating the need for positional supervision. First, we design a Relational Abstract Syntax Tree (RelAST), a generation-oriented representation that distills MathML trees into compact triplet sequences [S, R, D], where each token directly encodes a symbol identity, spatial relation, or nesting depth. Second, we introduce MathVAE, which learns structure-preserving latent representations through symbol-aware and relation-aware perceptual regularization, ensuring that the latent space captures both character semantics and spatial topology. Third, MathDiT performs conditional denoising in this structured latent space, further guided by a global symbol-count prior via Adaptive Layer Normalization (AdaLN) to improve structural coherence. Experiments show that DiffMath produces structurally consistent handwritten expressions, achieves superior performance over existing methods, and improves the accuracy of downstream OCR models through synthetic data augmentation.

Handwritten mathematical expression recognition (HMER) is conventionally scored by exact-match rates and string-similarity metrics that are blind to where an error occurs: two predictions with identical token-error counts receive identical scores whether they misplace a subscript or swap the operands of a fraction. Render-based character detection matching (CDM) aligns glyphs robustly but remains position-blind---on controlled fraction-operand swaps it scores 0.8595 where position-aware scoring yields 0.6253. Tree-edit metrics exhibit a complementary blind spot: rewrites outside the parser's normalization coverage are penalized as structural errors (0.8552 where render-based metrics score 1.0). We propose PA-CDM, a position-aware metric that couples character detection matching with position-forest encoding and divergence-level weighting; StructPerturb v2.0, a frozen benchmark of 1,340 controlled perturbation pairs across 15 type--intensity cells; and a cross-metric consistency protocol combining a sensitivity matrix, a human study, and LLM-judge calibration. In a six-annotator study, PA-CDM attains the highest correlation with human judgments among seven automatic metrics (Spearman rho=0.9535, n=990). A frontier LLM judge correlates slightly higher (rho=0.9613) but is costly, nondeterministic, and API-dependent; PA-CDM approaches it at zero marginal cost with deterministic, diagnosable behavior.

Non-Latin handwritten character recognition (HCR) remains understudied. Dominant methods consider it as generic image classification, which uses model scale to implicitly learn stroke structure. Structural-prior efficiency---the principle that explicitly encoding script-geometric regularities as architectural inductive biases can be both more accurate and require fewer parameters. We introduce GraphemeNet, a unified multi-script architecture, governed by two orthogonal binary axes. Axis 1 operationalises stroke-level geometric regularity via Persistent Scaffold Injection (PSI): a script-specific asymmetric convolution injects a stroke scaffold as a weighted residual at every encoder stage, continuously anchoring learned features to script geometry---distinct from skip connections, auxiliary losses, or attention reweighting. Axis 2 selects between global average pooling with gated fusion and cross-scale attention with a Stroke Topology Module (STM), depending on whether glyph discrimination requires spatial relational reasoning. A Linear Capsule Routing (LCR) with $O(n)$ routing is shared universally. On fourteen benchmarks across eight writing systems, the architecture generalises with only scaffold and decoder topology varying per script, consistently challenging, outperforming published baselines, and establishing structural-prior efficiency as a broadly applicable principle for multi-script HCR.