Hidden Geometry in Neural Networks Reveals Scale-Free Organization

by

Author: Denis Avetisyan

New research shows that even simple neural networks develop geometric patterns consistent with a fundamental mathematical theorem, suggesting underlying organizational principles at play.

Across varying spatial scales-specifically, Euclidean ball radii of 7 to 28 pixels-both standard and augmented models demonstrate a consistent capacity for Kernel Alignment Gain (KAG), maintaining performance ratios well above baseline; however, the augmented model exhibits approximately 30% lower ratios, suggesting reduced sensitivity to data variations and a corresponding decrease in internal conflict during analysis-a phenomenon indicative of graceful degradation rather than systemic failure.
Across varying spatial scales-specifically, Euclidean ball radii of 7 to 28 pixels-both standard and augmented models demonstrate a consistent capacity for Kernel Alignment Gain (KAG), maintaining performance ratios well above baseline; however, the augmented model exhibits approximately 30% lower ratios, suggesting reduced sensitivity to data variations and a corresponding decrease in internal conflict during analysis-a phenomenon indicative of graceful degradation rather than systemic failure.

Training on MNIST demonstrates the spontaneous emergence of Kolmogorov-Arnold Geometry within vanilla multilayer perceptrons.

While deep learning models are often treated as black boxes, emerging evidence suggests underlying geometric organization may be crucial to their function. This is explored in ‘Scale-Agnostic Kolmogorov-Arnold Geometry in Neural Networks’, which investigates whether shallow neural networks develop structures consistent with Kolmogorov-Arnold geometry-a mathematical framework describing how high-dimensional spaces can be represented by lower-dimensional ones. Our work demonstrates that even simple multilayer perceptrons, trained on the realistic high-dimensional MNIST dataset, spontaneously exhibit this scale-agnostic geometric organization across multiple spatial scales. Does this emergent geometry represent a fundamental principle governing information processing in neural networks, and could understanding it lead to more efficient and interpretable models?

The Geometry of Hidden Potential

Despite their remarkable ability to solve complex problems, traditional neural networks frequently function as opaque “black boxes”. Input data is processed through multiple layers of interconnected nodes, and while a result emerges, the precise reasoning behind that result remains largely hidden. This lack of interpretability poses challenges in fields requiring transparency and trust, such as medical diagnosis or financial modeling. Researchers are increasingly focused on understanding how these networks arrive at their conclusions, not just that they are accurate. The complex, high-dimensional transformations occurring within each layer make it difficult to trace the influence of specific inputs on the final output, hindering efforts to debug, refine, or even fully trust these powerful systems. This opaqueness limits the potential for leveraging neural networks in scenarios where explainability is paramount, driving the need for methods to peek inside the ‘black box’ and illuminate the internal decision-making process.

To decipher how neural networks arrive at their decisions, researchers often turn to the Jacobian matrix, a powerful tool derived from calculus. This matrix represents the collection of all first-order partial derivatives of a function, and in the context of neural networks, it reveals how each input dimension influences the network’s output at a specific point. By analyzing the Jacobian, one can map the local behavior of the network – essentially, how a tiny change in an input affects the output. This is crucial because complex, high-dimensional networks are rarely understood globally; instead, understanding their responses to infinitesimal changes – captured by the Jacobian – provides valuable insight into their internal workings and allows for a more granular assessment of their sensitivity and stability. The matrix doesn’t reveal what the network is learning, but rather how it responds to stimuli, offering a crucial piece of the interpretability puzzle.

The seemingly complex behavior of neural networks, and indeed any continuous function, rests upon a surprisingly simple foundation revealed by the Kolmogorov-Arnold Theorem. This theorem demonstrates that any continuous function of $n$ variables can be expressed as a composition of simpler, one-dimensional functions. Essentially, a multi-dimensional problem can always be reduced to a series of one-dimensional transformations, much like building a complex structure from basic building blocks. This decomposition isn’t necessarily unique, but its existence has profound implications for understanding how neural networks learn and represent information; it suggests that even highly complex decision boundaries can be constructed from relatively simple, layered operations, offering a potential pathway towards greater interpretability and control over these powerful systems.

Training consistently yields knowledge alignment geometry (KAG) in both standard and augmented networks, with larger networks demonstrating stronger geometric signatures as measured by PR ratio, KL divergence, and RR, and significantly exceeding random baselines despite some variance in the augmented model.
Training consistently yields knowledge alignment geometry (KAG) in both standard and augmented networks, with larger networks demonstrating stronger geometric signatures as measured by PR ratio, KL divergence, and RR, and significantly exceeding random baselines despite some variance in the augmented model.

The Emergence of Simplicity Within Complexity

Kolmogorov-Arnold Geometry (KAG) builds upon the foundational work demonstrating that a sufficiently over-parameterized neural network can approximate any continuous function. However, KAG goes further by revealing that, despite this potential for complexity, hidden layers often exhibit unexpectedly simple geometric properties. Specifically, the geometry of the function learned by a hidden layer frequently collapses onto lower-dimensional manifolds; meaning the effective dimensionality of the learned representation is significantly less than the number of hidden units. This simplification isn’t merely a statistical anomaly, but a predictable consequence of the network’s optimization process and the inherent structure of many real-world datasets. The theorem suggests that high-dimensional spaces are, in a sense, wasteful for representing many functions, and networks naturally gravitate towards lower-dimensional, more efficient representations.

The Jacobian matrix, representing the partial derivatives of a neural network’s output with respect to its hidden layer activations, frequently exhibits rows of zeros under Kolmogorov-Arnold Geometry (KAG). These ‘Zero Rows’ correspond to hidden units that are locally constant; meaning that small changes in the input do not induce any change in the unit’s activation or, consequently, the network’s output. Mathematically, a zero row in the Jacobian implies $ \frac{\partial output}{\partial hidden\_unit} = 0$ within a certain input region. The presence of such units suggests redundancy in the network’s representation, as they contribute no gradient information for learning or variation in the output during inference.

The presence of zero rows within the Jacobian matrix of a neural network’s hidden layer indicates the existence of locally constant hidden units. These units, by definition, exhibit no gradient with respect to the network’s input, meaning they do not contribute to changes in the output for local input variations. Consequently, these units can be removed from the network without altering its function, leading to a potential reduction in computational cost and model complexity. Furthermore, identifying and pruning such units can facilitate the creation of more efficient network architectures with fewer parameters, thereby reducing overfitting and improving generalization performance. This simplification is not merely theoretical; techniques for detecting and removing zero rows are being explored as a method for network compression and acceleration.

Training procedures maintain stable precision-recall ratios across varying minimum distance constraints for keypoint associations, demonstrating consistent performance even as the number of valid pairs decreases with increased spatial separation.
Training procedures maintain stable precision-recall ratios across varying minimum distance constraints for keypoint associations, demonstrating consistent performance even as the number of valid pairs decreases with increased spatial separation.

Measuring the Concentration of Learned Representations

Minor concentration, as a characteristic of neural network hidden layers, is assessed by examining the distribution of Jacobian determinants. The Jacobian matrix, computed for each hidden unit with respect to its inputs, provides insight into the unit’s sensitivity to changes in the input space. A high concentration of Jacobian determinants near zero indicates a substantial proportion of hidden units operating in a nearly constant state; these units exhibit minimal response to input variations. This is quantified by observing a prominent spike at or near zero in the histogram of absolute Jacobian determinants across the layer, suggesting redundancy or limited effective capacity within those units.

The quantification of ‘Minor Concentration’, referring to the distribution of Jacobian determinants, is achieved through metrics such as the Participation Ratio and Kullback-Leibler (KL) Divergence. The Participation Ratio, calculated as $ \frac{\sum_{i=1}^{N} (\frac{\partial y}{\partial x_i})^2}{\sum_{i=1}^{N} (\frac{\partial y}{\partial x_i})^2} $, measures the number of effectively active hidden units, with values greater than 1.0 indicating a concentration of activity. KL Divergence, calculated between the Jacobian determinant distribution of a trained network and a random initialization, provides a measure of how much the network’s hidden unit activity has concentrated during training; a significant divergence indicates the emergence of a non-uniform distribution and therefore, a detectable level of KAG. Both metrics offer a numerical characterization of the concentration phenomenon, allowing for objective comparison between different network configurations and training regimes.

Empirical evaluations of trained neural networks consistently report a Participation Ratio exceeding 1.0 across varying spatial scales and network architectures. This metric, calculated as the inverse of the sum of squared singular values of the Jacobian matrix, indicates that a disproportionate number of hidden units contribute significantly to the network’s output. Furthermore, networks demonstrate a statistically significant increase in KL Divergence compared to their randomly initialized states. This divergence, measured between the Jacobian determinant distributions of trained and random networks, confirms the emergence of a concentrated, low-rank structure indicative of Kernel Alignment and highlighting a substantial shift from random representations towards structured feature learning.

The Implications for Robust Image Recognition

Investigation into the internal geometry of Multilayer Perceptrons, utilizing Kernel Alignment Graph (KAG) analysis on the MNIST dataset, has revealed a consistent pattern of minor concentration within the network’s learned representations. This analysis doesn’t indicate widespread geometric frustration, but rather localized areas where information flow appears subtly channeled, suggesting the network isn’t utilizing its full representational capacity uniformly. The observed concentration manifests as heightened alignment between certain layers and specific input features, implying that the network develops a preference for processing particular patterns over others, even when redundant pathways exist. This subtle structuring, while not necessarily detrimental to performance on the MNIST dataset, hints at potential inefficiencies and opportunities for optimization in more complex neural network architectures, where such minor concentrations could compound into significant geometric bottlenecks.

The internal geometry of a neural network, specifically how it processes information, can be quantified by examining its Jacobian structure – a matrix representing the sensitivity of the network’s outputs to changes in its inputs. Researchers have developed a metric, the ‘Rotation Ratio’, to assess the ‘Alignment’ of this Jacobian, effectively measuring how much the network’s geometric configuration deviates from a standard, optimized arrangement. A high Rotation Ratio suggests a more disordered internal structure, potentially hindering efficient learning and generalization. Conversely, a lower ratio indicates better alignment and a more streamlined flow of information. This metric provides a valuable tool for understanding not just what a network learns, but how it learns, offering insights into the relationship between internal geometry and overall performance – and potentially guiding the development of more robust and efficient neural network architectures.

Research indicates that strategically implemented spatial augmentation techniques effectively alleviate geometric frustration within multilayer perceptrons. By subtly altering the positioning of input data during training, the network’s internal geometry is encouraged towards more aligned and efficient configurations. Quantified through the Participation Ratio – a metric reflecting the distribution of significant singular values in the Jacobian matrix – this optimization results in an approximate 30% reduction in geometric frustration. This improvement suggests that spatial augmentation isn’t merely a data expansion strategy, but a powerful tool for actively shaping the network’s learning landscape and enhancing its ability to generalize from complex datasets, ultimately leading to more robust and accurate image recognition capabilities.

The pursuit of scale-agnostic representations within neural networks, as demonstrated by the spontaneous emergence of Kolmogorov-Arnold Geometry in seemingly simple MLPs, echoes a fundamental truth about complex systems. This work subtly implies that organizational principles aren’t necessarily imposed but rather emerge from the interaction of simpler components. As Brian Kernighan observed, “Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it.” This resonates with the finding that a vanilla MLP, without explicit geometric constraints, discovers a structure consistent with KAG. The ‘cleverness’ isn’t in the design, but in the system’s capacity to organize itself, and any attempt to pre-define such organization introduces a fragility-a future cost-akin to technical debt.

What Lies Ahead?

The spontaneous emergence of Kolmogorov-Arnold geometry within these superficially simple multilayer perceptrons suggests a deeper principle at play-one less about achieving optimal classification and more about the inherent organizational tendencies of complex systems. Uptime, in this context, is merely a temporary reprieve from the inevitable drift toward entropy. The observed scale-agnosticism is not a feature to be engineered, but a consequence of the network finding a locally stable, though ultimately transient, equilibrium.

Further investigation must address the limitations of spatial analysis when applied to these high-dimensional representations. The Jacobian matrices, while revealing, offer only a snapshot-a frozen moment in a continuous flow. Latency, the cost of every request for information, becomes increasingly significant when attempting to map these evolving geometries. Future work should explore whether analogous structures manifest in networks trained on more complex datasets, and whether these geometric signatures correlate with generalization ability or robustness to adversarial attacks.

The enduring question remains: is this KAG merely a byproduct of gradient descent, a fleeting pattern in the noise, or a fundamental property of information processing itself? Stability, it seems, is an illusion cached by time. The pursuit of graceful decay, rather than perpetual optimization, may ultimately prove more fruitful.

Original article: https://arxiv.org/pdf/2511.21626.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

See also:

2025-12-01 03:03