Deep neural networks have shown many fruitful applications in this decade. A network can get the generalized function through training with a finite dataset. The degree of generalization is a realization of the proximity scale in the data space. Specifically, the scale is not clear if the dataset is complicated. Here we consider a network for the distribution estimation of the dataset. We show the estimation is unstable and the instability depends on the data density and training duration. We derive the kernel-balanced equation, which gives a short phenomenological description of the solution. The equation tells us the reason for the instability and the mechanism of the scale. The network outputs a local average of the dataset as a prediction and the scale of averaging is determined along the equation. The scale gradually decreases along training and finally results in instability in our case.

翻译：深度神经网络在近十年中展现出许多富有成效的应用。通过有限数据集的训练，网络能够获得泛化函数。泛化程度体现了数据空间中邻近尺度的实现。具体而言，当数据集复杂时，该尺度并不清晰。本文考虑用于数据集分布估计的网络。我们证明这种估计是不稳定的，且不稳定性取决于数据密度和训练时长。我们推导出核平衡方程，该方程给出了解的简短唯象描述。方程揭示了不稳定的原因和尺度的机制。网络输出数据集的局部平均值作为预测，而平均的尺度沿方程确定。在我们的案例中，该尺度随训练逐渐减小，最终导致不稳定性。