Rethinking the Harmonic Loss via Non-Euclidean Distance Layers

Neural networks trained with standard objectives exhibit behaviors characteristic of probabilistic inference: soft clustering, prototype specialization, and Bayesian uncertainty tracking. These phenomena appear across architectures -- in attention mechanisms, classification heads, and energy-based models -- yet existing explanations often rely on loose analogies to mixture models or post-hoc architectural interpretation. We provide a direct explanation. For any objective with log-sum-exp structure over distances or energies, the gradient with respect to each distance is exactly the negative posterior responsibility of the corresponding component: $\partial L / \partial d_j = -r_j$. The identity is algebraic, requiring only differentiability; it is a specialization of Fisher's identity, and its significance here is its address: standard neural objectives instantiate it without modification. The consequence is that gradient descent on such objectives performs generalized expectation-maximization implicitly, with responsibilities arising as gradients to be applied rather than auxiliary variables to be computed. This result unifies three regimes of learning: unsupervised mixture modeling, where responsibilities are fully latent; attention, where responsibilities are conditioned on queries; and cross-entropy classification, where supervision clamps responsibilities to targets. Our claims live at training time: the responsibility-weighted gradient dynamics recently documented in transformers follow from the objective's geometry. The in-context Bayesian computation that trained transformers perform at inference time is the endpoint of these dynamics, not their per-step content.

Large-scale Vision-Language Models (VLMs) such as CLIP learn powerful semantic representations but operate in Euclidean space, which fails to capture the inherent hierarchical structure of visual and linguistic concepts. Hyperbolic geometry, with its exponential volume growth, offers a principled alternative for embedding such hierarchies with low distortion. However, existing hyperbolic VLMs use entailment losses that are unstable: as parent embeddings contract toward the origin, their entailment cones widen toward a half-space, causing catastrophic cone collapse that destroys the intended hierarchy. Additionally, hierarchical evaluation of these models remains unreliable, being largely retrieval-based and correlation-based metrics and prone to taxonomy dependence and ambiguous negatives. To address these limitations, we propose an adaptive entailment loss paired with a norm regularizer that prevents cone collapse without heuristic aperture clipping. We further introduce an angle-based probabilistic entailment protocol (PEP) for evaluating hierarchical understanding, scored with AUC-ROC and Average Precision. This paper introduces a stronger hyperbolic VLM baseline ARGENT, Adaptive hieRarchical imaGe-tExt represeNTation. ARGENT improves the SOTA hyperbolic VLM by 0.7, 1.1, and 0.8 absolute points on image classification, text-to-image retrieval, and proposed hierarchical metrics, respectively.

Real-world data frequently exhibit latent hierarchical structures, which can be naturally represented by hyperbolic geometry. Although recent hyperbolic neural networks have demonstrated promising results, many existing architectures remain partially intrinsic, mixing Euclidean operations with hyperbolic ones or relying on extrinsic parameterizations. To address it, we propose the Intrinsic Lorentz Neural Network (ILNN), a fully intrinsic hyperbolic architecture that conducts all computations within the Lorentz model. At its core, the network introduces a novel point-to-hyperplane fully connected layer (FC), replacing traditional Euclidean affine logits with closed-form hyperbolic distances from features to learned Lorentz hyperplanes, thereby ensuring that the resulting geometric decision functions respect the inherent curvature. Around this fundamental layer, we design intrinsic modules: GyroLBN, a Lorentz batch normalization that couples gyro-centering with gyro-scaling, consistently outperforming both LBN and GyroBN while reducing training time. We additionally proposed a gyro-additive bias for the FC output, a Lorentz patch-concatenation operator that aligns the expected log-radius across feature blocks via a digamma-based scale, and a Lorentz dropout layer. Extensive experiments conducted on CIFAR-10/100 and two genomic benchmarks (TEB and GUE) illustrate that ILNN achieves state-of-the-art performance and computational cost among hyperbolic models and consistently surpasses strong Euclidean baselines. The code is available at magenta{this url}.