After the Euclidean Highway: Hyperbolic Expert AI as the Next Innovation

Expert domains are trees; the Euclidean transformer is not, diluting parent-child structure exponentially at depth. The hyperbolic turn left one question unasked: not how much of a network to curve, but where curvature may touch the gradient. Placement is a law, not a knob: the same geometry on a trainable adapter collapses training (seventeen training collapses, ~220 GPU-hours), yet at the loss layer alone it trains without one -- this is HySAT (Hyperbolic Structure-Aware Training), hyperbolic losses at the loss layer only. Across six expert SLMs we constructed and deployed (Llama 3.1 and EXAONE 3.5; four adapter strategies; 18.0M-sample corpus; zero NaN over ~317K optimizer steps), a matched four-arm ablation isolates the preserved manifold invariant, and three propositions and a lemma prove why loss-only placement is stable where adapter-on-manifold is not. Four models are operationally deployed (one live, consumer-facing), two open-weight, with per-step traces and a seventeen-incident failure ledger on Zenodo (CC-BY-4.0).

Enhancing Numerical Prediction in LLMs via Smooth MMD Alignment

Despite their strong general capabilities, large language models (LLMs) often remain unreliable when outputs must be numerically precise. A key reason is the training objective: standard cross-entropy treats numeric tokens as unstructured categories and ignores the metric structure of their values. We address this mismatch with Smooth Maximum Mean Discrepancy (SMMD), which builds on the classic MMD by incorporating value-distance kernels over numeric tokens and graph-based smoothness. With this kernel defined over a numeric sub-vocabulary, SMMD aligns the predicted numeric distribution to the target via kernel matching and smooths the prediction-target residual over the induced kernel graph to encourage local consistency. We evaluate SMMD on four numeric-target tasks: mathematical reasoning, arithmetic calculation, clock-time recognition, and chart question answering, across multiple open-weight LLM and VLM backbones. SMMD consistently improves accuracy over both cross-entropy and recent numeric-target losses; analyses show complementary effects between MMD and smoothness and underscore the importance of distance-based kernel design. Code is available at https://github.com/Zuozhuo/smmd-loss.

G-Loss: Graph-Guided Fine-Tuning of Language Models

Traditional loss functions, including cross-entropy, contrastive, triplet, and su pervised contrastive losses, used for fine-tuning pre-trained language models such as BERT, operate only within local neighborhoods and fail to account for the global semantic structure. We present G-Loss, a graph-guided loss function that incorporates semi-supervised label propagation to use structural relationships within the embedding manifold. G-Loss builds a document-similarity graph that captures global semantic relationships, thereby guiding the model to learn more discriminative and robust embeddings. We evaluate G-Loss on five benchmark datasets covering key downstream classification tasks: MR (sentiment analysis), R8 and R52 (topic categorization), Ohsumed (medical document classification), and 20NG (news categorization). In the majority of experimental setups, G-Loss converges faster and produces semantically coherent embedding spaces, resulting in higher classification accuracy than models fine-tuned with traditional loss functions.