Under the diagnostics, the audited formulations do not demonstrate an operative radial or cone-based hierarchy, and this report distills the audit into a five-number geometry report for evaluating future hierarchy claims.
Abstract
Hyperbolic vision-language models are designed to encode abstraction geometrically: general concepts near the origin, specific ones farther out, and entailment cones representing directed order. We ask whether trained MERU, HyCoCLIP, and PHyCLIP models actually use these mechanisms. We audit seven released checkpoints and matched from-scratch interventions, using diagnostics that distinguish active hyperbolic geometry from angular structure and supervision effects. All audited converged checkpoints remain near-Euclidean in the dimensionless radius $u=\sqrt{c}\rho$, which measures how strongly embeddings experience hyperbolic geometry: the largest observed image-side value is $0.37$ -- well below $u\approx0.84$, where local metric distortion reaches $10\%$. Releasing the curvature floor changes curvature and norms but not this regime, with mixed, generally modest downstream shifts. Trained entailment cones are saturated or nearly saturated, so low violation rates can arise from trivially wide cones rather than learned order. Preregistered semantic traversal detects weak within-branch order but no operative full-hierarchy readout. Shuffle-controlled tests detect no pair-specific radial ordering in released checkpoints, and no positive result is consistent across all three matched ViT-B seeds. We trace this to a low-curvature shortcut: lowering curvature widens entailment cones and suppresses violations without learning order. In the probed trajectories, gradient decomposition identifies entailment as the dominant curvature-lowering pressure during collapse. Yet curvature contracts even when entailment is removed, so the shortcut is not the sole cause. Under our diagnostics, the audited formulations do not demonstrate an operative radial or cone-based hierarchy. We distill the audit into a five-number geometry report for evaluating future hierarchy claims.
Vision language models face a fundamental geometry trade-off: Euclidean representations excel at instance-level discrimination, while hyperbolic representations naturally encode semantic hierarchies. Hybrid training is challenging because one geometry may dominate early, leaving the other under-trained failure mode we term geometry dominance. We introduce Adaptive Geometry Routing (AGR), a framework that addresses this via a novel four-phase training curriculum : (1) Isolation hyperbolic-only training stabilizes hierarchical structure; (2) Shadow router learns mixing patterns using only hyperbolic signals; (3) Soft Launch Euclidean scores gradually become visible; (4) Adaptive full dual-geometry routing. This phased coordination of router activation (?) and Euclidean visibility (?) prevents early dominance while enabling data-driven geometry selection. Built on a shared backbone with lightweight LoRA-adapted heads and bounded residual corrections, AGR discovers that hyperbolic geometry is preferred by default (85% weight), with routing adapting semantically abstract queries route more hyperbolic, attribute-rich queries shift toward Euclidean. On ViT-B, AGR achieves 38.8% COCO T2I R@5 (+6.5pp over MERU), 64.7% Flickr30K I2T R@5 (+11.3pp), and 32.6% ImageNet accuracy (+9.3pp), demonstrating that phased curriculum training enables stable hybrid geometry learning for vision-language understanding.
Sarthak Srivastava, Kathy Wu· Proceedings of the 32nd ACM...· 0 citations
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).
Vision-Language-Action (VLA) models encode visual observations as flat 2D patch tokens that carry no intrinsic geometric structure, and augmenting them with dense monocular depth injects per-pixel scalar values that encode neither surface orientation nor geometric confidence. This leaves the policy with limited structured spatial reasoning for action prediction. We propose GaussVLA, a Mamba-based VLA that incorporates two custom modules: Gaussian Spatial Tokenizer (GST) to lift frozen semantic and depth features into compact 3D Gaussian tokens, pools geometrically salient regions with learned queries, and \emph{Depth-Aware Chain-of-Thought (DA-CoT)} that performs structured, non-autoregressive geometric reasoning under language and flow-time conditioning. Across both simulation and real-world evaluations, GaussVLA demonstrates strong spatial-manipulation performance while remaining parameter-efficient. On LIBERO, it achieves 93.5% average success and 100.0% success on the Spatial suite with only 200M parameters, improving over SpatialVLA by 19.7% relative average success while remaining significantly more parameter-efficient.
MD SELIM SAROWAR, Md Tanvir Islam, Sungho Kim et al.· 0 citations
Kolmogorov-Arnold Networks (KANs) replace fixed activations in deep architectures with learnable univariate edge functions, making the choice of edge parametrisation central. Existing variants rely on fixed bases such as splines, polynomials, or Fourier features, which impose a function-space geometry before data are observed. We introduce geometry-constrained KANs, a family of edge activations derived from Banach duality maps in which the geometry itself is learned through a scalar exponent $p>1$ per edge. This exponent controls the qualitative response: sub-Euclidean values produce sharp, threshold-like behaviour reminiscent of the $\ell_1$ (LASSO) geometry, $p = 2$ recovers the linear regime, and larger values produce flatter responses near the origin. Across 50 symbolic-regression targets ($40$ from the AI Feynman benchmark plus $10$ synthetic stress tests), geometry-constrained KANs match or beat every fixed-basis baseline on median NRMSE (Banach-KAN $0.030$, tying Chebyshev and improving on splines); on average rank Banach-KAN is best on the $18$-equation core ($2.00$) and statistically tied with the strongest spline on the full benchmark ($2.32$ vs. $2.34$). The clearest gains appear under measurement noise: as $\sigma$ grows from $0$ to $1$, $\ell^p$-KAN degrades only $3.7\times$ -- below even a cross-validated spline ($\approx 11\times$) -- while an unregularised spline degrades $21.6\times$; Banach-KAN degrades $8.8\times$, comparable to a tuned spline but far more stable than the unregularised one. Banach-KAN also takes the most per-equation wins in the small-sample regime, with fixed-basis models catching up only as the training set grows. Learned exponents provide an interpretable, relative signal: at a fixed initialisation they reveal a consistent, target-dependent geometric ordering across equation families and input dimensions.
We identify a recurrent algebraic regularity in Transformer attention: a sparse subset of effective OV operators $T=OV^\top$ nearly closes under composition, $T^2\approx\alpha T$. Across six pretrained endpoints spanning 2.8B--235B parameters, 3.98--8.00% of heads reach squared closure alignment $\mathcal{P}\geq0.9$, while no matched within-layer O/V mismatch does. An exact principal-coordinate factorization, $T=Q_OKQ_V^\top$ and $T^2=Q_O(KDK)Q_V^\top$, separates within-support transport from read--write return geometry. Across all 7,304 heads in nine MHA/GQA models, scrambling only the orientation of $K$ while preserving singular values, norms, factor spans, and principal angles reduces median closure from 0.336 to $1.04\times10^{-4}$; trained orientation wins for 98.64% of heads and in every layer. Constructive searches show that high closure is feasible in every surveyed layer, but usually not attained. Retrospective trajectories in three independently trained lineages further separate broadly available capacity from the orientations attained by final strong heads. Under exact value sharing, headwise closure extends to a right-action algebra, $T_iT_j=\alpha_jT_i$. Seven-model experiments verify the approximate law and reveal distinct oblique projections with a shared value-defined kernel. These results characterize scaled idempotence as a sparse trained orientation within broadly available geometric capacity and show how value sharing extends a headwise relation into a local operator algebra.
This work presents a systematic scaling law study for text-to-image diffusion models using Abra, a controlled family of flow-matching transformers trained across three orders of magnitude worth of compute, demonstrating that diffusion models scale just as predictably as language models but require far more data to train optimally.
Kyle R. Chickering, Wei-An Lin, Swayam Bhanded et al.· 1 citation
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