LGQ: Learning Discretization Geometry for Scalable and Stable Image Tokenization

Discrete image tokenization is a key bottleneck for scalable visual generation: a tokenizer must remain compact for efficient latent-space priors while preserving semantic structure and using discrete capacity effectively. Existing quantizers face a trade-off: vector-quantized tokenizers learn flexible geometries but often suffer from biased straight-through optimization, codebook under-utilization, and representation collapse at large vocabularies. Structured scalar or implicit tokenizers ensure stable, near-complete utilization by design, yet rely on fixed discretization geometries that may allocate capacity inefficiently under heterogeneous latent statistics. We introduce Learnable Geometric Quantization (LGQ), a discrete image tokenizer that learns discretization geometry end-to-end. LGQ replaces hard nearest-neighbor lookup with temperature-controlled soft assignments, enabling fully differentiable training while recovering hard assignments at inference. The assignments correspond to posterior responsibilities of an isotropic Gaussian mixture and minimize a variational free-energy objective, provably converging to nearest-neighbor quantization in the low-temperature limit. LGQ combines a token-level peakedness regularizer with a global usage regularizer to encourage confident yet balanced code utilization without imposing rigid grids. Under a controlled VQGAN-style backbone on ImageNet across multiple vocabulary sizes, LGQ achieves stable optimization and balanced utilization. At 16K codebook size, LGQ improves rFID by 11.88% over FSQ while using 49.96% fewer active codes, and improves rFID by 6.06% over SimVQ with 49.45% lower effective representation rate, achieving comparable fidelity with substantially fewer active entries. Our GitHub repository is available at: https://github.com/KurbanIntelligenceLab/LGQ

Multivariate de Bruijn Graphs: A Symbolic Graph Framework for Time Series Forecasting

Time series forecasting remains a challenging task for foundation models due to temporal heterogeneity, high dimensionality, and the lack of inherent symbolic structure. In this work, we propose DRAGON (Discrete Representation and Augmented Graph encoding Over deBruijN Graphs), a novel encoder that introduces Multivariate de Bruijn Graphs (MdBGs) to bridge the gap between symbolic representations and neural modeling. DRAGON discretizes continuous input sequences and maps them onto a fixed graph structure, enabling dynamic context recovery via graph-based attention. Integrated as an auxiliary module within a dual-branch architecture, DRAGON augments conventional CNN-based encoders with symbolic, structure-aware representations. All code developed for this study is available at: https://github.com/KurbanIntelligenceLab/MultdBG-Time-Series-Library