Scale-Agnostic Kolmogorov-Arnold Geometry in Neural Networks
This addresses the problem of understanding geometric structure in neural networks for machine learning researchers, but it is incremental as it extends prior synthetic findings to a more realistic dataset.
The paper investigated whether Kolmogorov-Arnold geometric structure emerges in neural networks on realistic high-dimensional data, finding that it appears consistently across spatial scales in MNIST digit classification tasks.
Recent work by Freedman and Mulligan demonstrated that shallow multilayer perceptrons spontaneously develop Kolmogorov-Arnold geometric (KAG) structure during training on synthetic three-dimensional tasks. However, it remained unclear whether this phenomenon persists in realistic high-dimensional settings and what spatial properties this geometry exhibits. We extend KAG analysis to MNIST digit classification (784 dimensions) using 2-layer MLPs with systematic spatial analysis at multiple scales. We find that KAG emerges during training and appears consistently across spatial scales, from local 7-pixel neighborhoods to the full 28x28 image. This scale-agnostic property holds across different training procedures: both standard training and training with spatial augmentation produce the same qualitative pattern. These findings reveal that neural networks spontaneously develop organized, scale-invariant geometric structure during learning on realistic high-dimensional data.