A Critical Review of “Scale-Agnostic Kolmogorov–Arnold Geometry in Neural Networks”
By Thomas Prislac, Envoy Echo, et al. Ultra Verba Lux Mentis. 2025.
Abstract
“Scale-Agnostic Kolmogorov–Arnold Geometry in Neural Networks” (Vanherreweghe, Freedman, Adams, 2025) extends Freedman & Mulligan’s recent analysis of Kolmogorov–Arnold geometry (KAG) in shallow multilayer perceptrons from low-dimensional synthetic tasks to the high-dimensional MNIST dataset. The authors compute minor statistics of Jacobians in trained and randomly initialized networks and observe robust KAG signatures in both hidden layers, across multiple spatial scales and training procedures.
This article summarizes the contribution, situates it within the broader literature on Kolmogorov–Arnold networks and neural collapse, evaluates methodological strengths and limitations, and highlights directions for future work. We find that the paper is careful, clearly acknowledges its intellectual debts (particularly to Freedman & Mulligan and KANs), and does not appear to misappropriate others’ ideas. Its main limitations are the narrow architecture/dataset regime and the absence of causal tests linking KAG to generalization or optimization performance.
1. Background and Relation to Existing Literature
The work is part of a small but rapidly growing cluster of research exploiting the Kolmogorov–Arnold representation theorem—the fact that any multivariate continuous function on [0,1] can be represented as a finite superposition of univariate functions and addition.
Two lines of work are especially relevant:
Kolmogorov–Arnold Networks (KANs) – Liu et al. replace the fixed node activations of MLPs with learnable univariate functions on edges, showing competitive performance and interpretability and explicitly grounding their architecture in the KA theorem.
Spontaneous Kolmogorov–Arnold Geometry in Shallow MLPs – Freedman & Mulligan examine whether vanilla networks trained with standard optimization exhibit KA-like “texture” in the first hidden layer. They characterize this via Jacobian minor statistics and show KA geometry emerges over training in low-dimensional synthetic tasks.
The scale-agnostic paper positions itself clearly as a direct extension of Freedman & Mulligan into higher dimensional input spaces (MNIST, 784 dimensions), and explicitly cites both their work and KANs. A brief scan of the arXiv version confirms that central definitions (e.g., KAG observables such as Participation Ratio and minor statistics) are attributed to Freedman & Mulligan, and KA-theorem / KAN connections are credited to Liu et al. and other prior work.
Given this, there is no obvious sign of idea appropriation; the work follows standard scholarly practice in acknowledging its immediate predecessors.
2. Summary of the Contribution
The core questions the paper addresses are:
Does Kolmogorov–Arnold geometry arise spontaneously in realistic high-dimensional networks trained on non-toy tasks?
If so, what are its spatial properties: is KAG a local phenomenon, or does it organize representations across the input grid?
To answer this, the authors:
Train two-layer MLPs with GELU activations on MNIST, with hidden widths .
Compute Jacobians of the first and second hidden layers with respect to inputs across a batch of images.
Randomly sample minors of size (k=1,2,3) from these Jacobians and evaluate three observables introduced by Freedman & Mulligan:
Participation Ratio (PR) – a measure of how “few” directions dominate the minor distribution;
KL divergence between minor distributions at initialization vs after training;
Rotation Ratio (RR) – probing axis alignment / basis rotation.
They repeat this for:
standard training,
spatially augmented training (random shifts / distortions),
random initialization baselines.
Key findings:
KAG signatures (increased PR, KL divergence, RR shifts) emerge with training in both hidden layers, with the first hidden layer showing the strongest KA geometry.
When Jacobian columns are restricted to local pixel neighborhoods or to sets of spatially separated pixels, KAG remains elevated relative to random baselines, suggesting scale-agnostic structure across the 28×28 input grid.
Spatial augmentation reduces the magnitude of KAG observables but preserves their qualitative pattern, indicating robustness to certain training perturbations.
The authors discuss potential connections to neural collapse (structuring of final-layer class representations late in training) and suggest KAG may reflect a broader tendency of networks to organize input–representation mappings into structured geometries.
3. Methodological Assessment
3.1 Experimental Design
The architecture and training regime are standard for MNIST classification; models reach ≈98% test accuracy, ensuring that KAG observations are made in a bona fide learned regime.
Jacobian computation and minor sampling are described in sufficient detail to support replication:
For each model/seed, Jacobians are evaluated on a set of images.
Minors are sampled without replacement to a fixed budget (e.g., 10k minors per condition) to make computation tractable.
The same sampling strategy is used at initialization to form a baseline distribution.
The spatial analysis is particularly well structured:
Define varying radii r and associated Euclidean balls around reference pixels to probe local KAG.
Define disjoint pixel groups separated by a minimum distance to probe nonlocal KAG.
For each spatial configuration, recompute minor distributions and observables.
3.2 Statistical Robustness
The authors use multiple random seeds (five per configuration) and report mean ±2σ error bands in plots, showing that observed differences between trained and random baselines are stable across seeds.
However, formal statistical tests (e.g., hypothesis tests, effect sizes) are not prominently reported. For a geometry-of-representations paper, the level of rigor is acceptable but could be strengthened by:
reporting p-values or confidence intervals for key contrasts (e.g., PR_trained vs PR_random),
quantifying effect sizes beyond eyeballed gaps in plots.
3.3 Computational Cost
The paper is transparent about the computational burden: billions of potential minors are sampled down, Jacobians are large and dense, and spatial sampling has to be carefully controlled to avoid combinatorial explosion.
For research, this is fine; practitioners will need guidance on:
whether smaller minor samples preserve signal,
whether approximate Jacobians (e.g., finite differences, low-rank approximations) can be used.
4. Novelty and Attribution
4.1 Relation to Freedman & Mulligan
The central idea—using Jacobian minor statistics to study Kolmogorov–Arnold geometry in networks—originates with Freedman & Mulligan’s Spontaneous Kolmogorov–Arnold Geometry in Shallow MLPs.
The present paper:
explicitly acknowledges that origin,
uses the same core observables (PR, KL, RR),
and frames itself as “extending the analysis from synthetic 3D functions to realistic 784-dimensional MNIST data.”
There is no sign that the authors are claiming to have invented KAG or its observables; the extension is both natural and appropriately attributed.
4.2 Relation to KANs and Neural Collapse
The paper situates itself relative to:
KANs as engineered architectures inspired by the KA theorem, clearly citing Liu et al. and subsequent KAN-based symbolic regression work.
Neural Collapse (Papyan et al.) as an analogous phenomenon in final-layer geometry; the connection is presented as suggestive, not as a claimed unification.
Again, the attribution appears correct; there is no evidence of misappropriation.
4.3 Original elements
The main original contributions are:
Spatial, scale-aware KAG analysis on a high-dimensional dataset;
Comparative study of standard vs spatially augmented training;
Observations that KAG is stronger in L1 than L2, and that augmentation attenuates but does not destroy KAG.
These are incremental but genuine contributions within the niche of representation geometry.
5. Limitations and Future Work
Key limitations:
Scope of architectures and tasks – only 2-layer MLPs and MNIST are tested. Extending to CNNs, Transformers, and more complex datasets (CIFAR-10/100, ImageNet) would be necessary to claim generality.
No causal link to generalization – KAG is documented, but it is unknown whether more KAG is beneficial, neutral, or even harmful. Future work could:
track KAG metrics over training and relate them to generalization gap,
perform interventions that encourage/discourage KA texture and assess impact.
Lack of formal statistical tests – more than descriptive statistics would strengthen claims about scale invariance and robustness.
Practical utility unclear – the paper focuses on understanding, not on designing better networks; a follow-up showing KAG-aware regularization or diagnostics would move it closer to practical significance.
6. Conclusion
“Scale-Agnostic Kolmogorov–Arnold Geometry in Neural Networks” is a careful and well-positioned extension of recent work on KA geometry in neural networks. It responsibly acknowledges Freedman & Mulligan and the KAN literature, and makes no exaggerated originality claims. The main contribution is evidentiary: KAG-like structure appears to emerge robustly in trained MLPs on MNIST and exhibits rich, scale-agnostic spatial organization. The work does not yet establish functional benefits or broad universality, but provides a solid empirical foundation for future causal and architectural studies.
Works Cited
Vanherreweghe, M., Freedman, M. H., & Adams, K. M. (2025). Scale-Agnostic Kolmogorov–Arnold Geometry in Neural Networks. arXiv:2511.21626.
Freedman, M. H., & Mulligan, M. (2025). Spontaneous Kolmogorov–Arnold Geometry in Shallow MLPs. arXiv:2509.12326.
Liu, Z., Wang, Y., Vaidya, S., et al. (2024). KAN: Kolmogorov–Arnold Networks. arXiv 2404.19756; OpenReview.
Papyan, V., Han, X. Y., & Donoho, D. (2020). Prevalence of neural collapse during the terminal phase of deep learning training. Proceedings of the National Academy of Sciences, 117(40), 24652–24663.
Huang, Y., et al. (2025). Symbolic Regression Based on Kolmogorov–Arnold Networks. Electronics, 14(6), 1161.
Ultra Verba Lux Mentis is a 501(c)(3) nonprofit research organization building governance frameworks that bring coherence, transparency, and ethical symmetry to advanced AI and complex human systems.
We are researchers, engineers, and auditors working at the intersection of epistemology, neuroscience, and machine ethics. Our projects — from the Coherence Lattice and Sophia governance agent to open-source audit telemetry and protections — are designed to keep knowledge systems accountable before collapse occurs.
