神经网络中的尺度无关的柯莫戈罗夫-阿诺尔德几何
Scale-Agnostic Kolmogorov-Arnold Geometry in Neural Networks
AI总结：
研究显示神经网络在高维数据学习中自发形成尺度无关的柯莫戈罗夫-阿诺尔德几何结构。
摘要：
弗里德曼和穆拉加恩最近的研究表明，在合成三维任务上训练浅层多层感知机时，会自发地发展出柯莫戈罗夫-阿诺尔德几何（KAG）结构。然而，这种现象在现实的高维设置中是否持续存在，以及这种几何结构表现出什么样的空间属性仍然不清楚。我们通过使用具有系统空间分析的2层MLP扩展KAG分析，应用于MNIST数字分类（784维）。我们发现KAG在训练过程中出现，并在不同空间尺度上一致存在，从局部的7像素邻居到完整的28x28图像。这种尺度无关的特性在不同的训练过程下保持不变：无论是标准训练还是带有空间增强的训练，都产生相同的结果。这些发现表明，在现实的高维数据上学习时，神经网络会自发地发展出组织良好且尺度不变的几何结构。
英文摘要：
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.
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